Penn Institute for Economic Research
Department of Economics
University of Pennsylvania
3718 Locust Walk
Philadelphia, PA 19104-6297
pier@econ.upenn.edu
http://economics.sas.upenn.edu/pier
PIER Working Paper 13-042
“Modeling the Evolution of Expectations and
Uncertainty in General Equilibrium”
by
Francesco Bianchi and Leonardo Melosi
http://ssrn.com/abstract=2275515
Modeling the Evolution of Expectations
and Uncertainty in General Equilibrium
Francesco Bianchi
Duke University
Leonardo Melosi
Federal Reserve Bank of Chicago
December, 2012
Abstract
This paper develops methods to study the evolution of agents’ expectations and uncertainty in
general equilibrium models. A central insight consists of recognizing that the evolution of agents’
beliefs can be captured by de…ning a set of regimes that are characterized by the degree of agents’
pessimism, optimism, and uncertainty about future equilibrium outcomes. Once this kind of structure
is imposed, it is possible to create a mapping between the evolution of agents’beliefs and observable
outcomes. Agents in the model are fully rational, conduct Bayesian learning, and they know that they
do not know. Therefore, agents form expectations taking into account that their beliefs will evolve
according to what they observe in the future. The new modeling framework accommodates both
gradual and abrupt changes in agents’beliefs and allows an analytical characterization of uncertainty.
Shocks to beliefs are shown to have both …rst-order and second-order e¤ects. To illustrate how to
apply the methods, we use a prototypical Real Business Cycle model in which households form beliefs
about the likely duration of high-growth and low-growth regimes.
Keywords: Markov switching DSGE model, Bayesian econometrics, beliefs, uncertainty, Bayesian
learning, rational expectations.
JEL classi…cation: D83, E52, E52.
We thank Martin Eichenbaum, Eric Leeper, Dan Waggoner, and Tao Zha for very useful comments and
discussion. Todd Messer provided excellent research assistance. Francesco Bianchi gratefully acknowledges …nancial support from the National Science Foundation through grant SES-1227397. Francesco Bianchi, Department of Economics, Duke University, 213 Social Sciences Building, Durham, NC, 27708-0089, USA. Leonardo
Melosi, Federal Reserve Bank of Chicago, 230 South La Salle street, Chicago, IL 60604-1413, USA. Email:
lmelosi@frbchi.edu. The views in this paper are solely the responsibility of the authors and should not be
interpreted as re‡ecting the views of the Federal Reserve Bank of Chicago or any other person associated with
the Federal Reserve System.
1
Introduction
A centerpiece of the rational expectations revolution is that macroeconomic outcomes critically
depend on how agents’expectations about future events evolve over time. Therefore, correctly
modeling the dynamics of the private sector’s beliefs is essential to accurately predict economic
outcomes. Most general equilibrium models are solved assuming that agents know the exact
structure of the whole economy and are certain about the rules governing the future behavior
of other players. This is certainly a strong restriction imposed upon the dynamics of beliefs.
For instance, the private sector is likely to have limited information about the future path of
policy-makers’decisions, dividend payments, economic growth, etc.
In this paper we develop methods to solve models in which forward-looking and fully rational
agents are subject to waves of pessimism, optimism, and uncertainty that turn out to critically
a¤ect macroeconomic outcomes. Such outbursts of pessimism, optimism, and uncertainty may
happen abruptly or may gradually unfold over a long period of time in response to the behavior
of other agents or to the realizations of economic outcomes. All results are derived within a
modeling framework suitable for structural estimation that will allow researchers to bring the
models to the data.
The evolution of agents’beliefs is modelled assuming the existence of di¤erent states of the
world that di¤er according to the statistical properties of the exogenous shocks or based on the
behavior of some of the agents in the model. Such regimes follow a Markov-switching process,
which may be correlated with other aspects of the model. For example, the government could
be more likely to in‡ate debt away when the level of spending is high. Agents are assumed to
observe economic outcomes, but not the regimes themselves. Agents will then adopt Bayesian
learning to infer which regime is in place. This will determine the evolution of agents’beliefs
about future economic outcomes.
Our modeling framework goes beyond the assumption of anticipated utility that is often
used in models characterized by a learning process. Such an assumption implies that agents
forecast future events assuming that their beliefs will never change in the future. Instead, agents
in our models know that they do not know. Therefore, when forming expectations, they take
into account that their beliefs will evolve according to what they observe in the future. In our
context, it is possible to go beyond the anticipated utility assumption because there are only
a …nite number of relevant beliefs and they are strictly linked to observable outcomes through
the learning mechanism in a way that we can keep track of their evolution.
The proposed model framework is ‡exible enough to encompass both abrupt and gradual
changes in beliefs. For example, augmenting the modeling framework with signals about the
regime in place allows one to capture the e¤ect of news on the evolution of the economy or
to study the macroeconomic implications of changes in animal spirits about future events. At
1
the same time, through the learning process, we can model situations in which agents’beliefs
gradually change in response to the behavior of other agents or the realizations of stochastic
events. This sluggish adjustment of public expectations is hard to reproduce through rational
expectations models in which the functioning of the whole economy is common knowledge
among agents.
We show how to apply these methods using a prototypical Real Business Cycle (RBC)
model. In the model, total factor productivity (TFP) growth can assume two values: high or
low. For each value of TFP growth, we allow for a long-lasting and a short-lasting regime.
Therefore, while agents can observe the current TFP growth rate, they are uncertain about
its future values, because they do not know if the current value is likely to last for a short
time or for a long time. We consider a wide range of speci…cations, allowing for smooth
transitions or abrupt changes in agents’ optimism about future realizations of TFP growth.
Each of these di¤erent speci…cations can be easily captured with the appropriate transition
matrix governing the evolution of TFP growth. This has the important implication that the
dynamics of pessimism, optimism, and uncertainty are consistent in equilibrium. Whenever a
short-lasting regime is in fact realized, with the bene…t of hindsight, agents’beliefs turn out to
over-react to the regime change because agents always take into account the possibility that the
economy entered a long-lasting regime. On the other hand, if, in fact, the regime is long-lasting,
it takes time for agents’beliefs to line up with the actual realization. This implies that although
agents are fully rational, their beliefs are generally misaligned with respect to the actual state of
the economy. Such a misalignment is found to substantially in‡uence consumption and capital
allocation in the RBC model.
The methods introduced in this paper can be combined with the techniques developed
by Bianchi (2012) to obtain an analytical characterization of the evolution of uncertainty in
response to changes in agents’ beliefs. We expand our analysis of the RBC model to study
the case in which agents receive signals about the likely duration of the current regime. In
this environment, signals work as shocks to agents’ beliefs that have …rst-order and secondorder e¤ects. Uncertainty about macroeconomic events evolves over time as agents’ beliefs
drift, creating interesting comovements between volatility and real activity. This might shed
further light on the link between uncertainty and macroeconomic outcomes with respect to the
fascinating study of Bloom (2009).
The methods developed in this paper are based on the idea of expanding the number of
regimes to take into account the learning mechanism. The central insight consists of recognizing
that the evolution of agents’ beliefs can be captured by de…ning an expanded set of regimes
indexed with respect to agents’ beliefs themselves. Once this structure has been imposed,
the model can be recast as a Markov-switching dynamic stochastic general equilibrium (MSDSGE) model with perfect information. If regime changes enter additively the model can be
2
solved with standard solution methods such as gensys (Sims, 2002) and Blanchard and Kahn
(1980), following the approach described in Schorfheide (2005) and Liu, Waggoner, and Zha
(2011). If instead regime changes enter multiplicatively the model can be solved with any of the
methods developed for solving MS-DSGE models, such as Davig and Leeper (2007), Farmer,
Waggoner, and Zha (2009), Cho (2012), and Foerster, Rubio-Ramirez, Waggoner, and Zha
(2011).
In both cases, the resulting solution is suitable for likelihood-based estimation. This is
because even if the …nal number of regimes is very large, there is a tight link between observable
outcomes and the evolution of agents’beliefs. In other words, the transition matrix governing
the joint dynamics of the economy and agents’beliefs is highly restricted. For example, Bianchi
and Melosi (2012a) apply these methods and Bayesian techniques to estimate a model in which
agents are uncertain about the future stance of monetary policy. This paper is therefore related
to a growing literature that models parameter instability to capture changes in the evolution of
the macroeconomy. This consists of two branches: Schorfheide (2005), Justiniano and Primiceri
(2008), Bianchi (forthcoming), Davig and Doh (2008), and Fernandez-Villaverde and RubioRamirez (2008) introduce parameter instability in DSGE models, while Sims and Zha (2006),
Primiceri (2005), and Cogley and Sargent (2005) work with structural VARs. Finally, to the
extent that we can model situations in which agents’beliefs evolve in response to policy-makers’
behavior, our work is also linked to papers that study how in‡ation expectations respond to
policy decisions, such as Mankiw, Reis, and Wolfers (2004), Nimark (2008), Del Negro and
Eusepi (2010), and Melosi (2012a,b).
The remainder of the paper is organized as follows. Section 2 introduces the class of models
and derives the main results. Section 3 applies the methods to a prototypical RBC model in
which agents try to infer the likely value of future TFP growth. Section 4 extends the analysis
to allow for signals. Section 5 provides an overview of alternative applications. Section 6
concludes.
2
The Model Framework
In this section, we introduce the modeling environment to which our methods are applicable.
This framework turns out to be suitable for studying the e¤ects of waves of pessimism, optimism,
and uncertainty on macroeconomic dynamics. Various sources of uncertainty can be considered,
such as dividends, relative volatility of shocks, or the type of the central bank. For instance, one
can use this framework to study how animal spirits about the incoming phase of the business
cycle evolve over time and, in turn, a¤ect the business cycle itself. Alternatively, one can
use this framework to study how long it takes agents to learn about structural changes in the
dynamics of fundamentals or in the behavior of policy-makers.
3
The class of models we focus on has three salient features:
1. A system of equations for the variables of the model:
0
( t ) St =
c
( t) +
1
( t ) St
1
+
( t ) "t +
t
(1)
where St is a vector containing all variables of the model known at time t (including
conditional expectations formed at time t), t is a vector containing the endogenous expectation errors, and the random vector "t contains the familiar Gaussian shocks. The
hidden variable t controls the parameter values in place at time, ( t ) ; assumes discrete values t 2 f1; : : : ; ng, and evolves according to a Markov-switching process with
transition matrix P.
2. Agents have to forecast the dynamics of the endogenous variables St+1 on the basis of
Model (1) and their information set at time t, It . This includes the history of model
variables and shocks, but not the history of regimes, t : It fS t ; "t g :
3. Some regimes are assumed to bring about the same model parameters, ( t ). Let us
group the regimes into m blocks bj = f t 2 f1; : : : ; ng : ( t ) = bi g, for j 2 f1; :::; mg.
Given that agents know the structure of the model (sub 1 ) and can observe the endogenous
variables and the shocks (sub 2 ), they can also determine which set of parameters is in place
at each point in time. However, while this is enough to establish the history of blocks, agents
cannot exactly infer the realized regime t ; because the regimes within each block share the
same parameter values (sub 3 ). It is very important to emphasize that regimes that belong to
the same block are not identical in all respects, as they can di¤er in their stochastic properties
such as average persistence and the probability of switching to other regimes. These properties
are known to agents that will use them to learn about the regime in place today and to form
expectations about the future. Therefore, points 1-3 describe a model in which agents learn
about the latent variable t . As will be shown below, such a learning process a¤ects the
equilibrium law of motion of the economy. However, agents cannot extract any additional
information about the underlying regime from observing the resulting law of motion because
this re‡ects their own beliefs.
Henceforth, we will consider a benchmark case in which there are two blocks (m = 2) and
two regimes within each block. This choice is made in order to keep notation simple. The
extension to the case in which m > 2 is straightforward. The probabilities of moving across
4
regimes are summarized by the transition matrix:
2
p11
6
6 p21
P =6
6 p
4 31
p41
p12
p22
p13
p23
p32
p42
p33
p43
3
p14
7
p24 7
7
p34 7
5
p44
(2)
in which the probability of switching to regime j given that we are in regime i is denoted by
pij . Without loss of generality, we assume that regimes t = 1 and t = 2 belong to Block 1,
while regimes t = 3 and t = 4 belong to Block 2. We consider only non-trivial blocks that
satisfy p11 + p12 + p21 + p22 6= 0 and p33 + p34 + p43 + p44 6= 0. The excluded cases are trivial
as both blocks last only one period. Furthermore, we require that the two regimes that belong
to the same block di¤er either in their persistence or in the probability of moving from one
another; that is, we require that either p11 6= p22 or p12 6= p21 and either p33 6= p44 or p34 6= p43 .
This condition makes the within-block Bayesian learning non-trivial. Finally, we will impose
that p11 + p22 > 0 and p33 + p44 > 0. This last assumption guarantees that within a block at
least one of the two regimes can last more than one period. Summarizing, for each block, we
will maintain the following benchmark assumptions throughout the paper:
A1 Non-triviality assumption: p11 + p12 + p21 + p22 6= 0 and p33 + p34 + p43 + p44 6= 0
A2 Non-trivial-learning assumption: Either p11 6= p22 or p12 6= p21 and either p33 6= p44 or
p34 6= p43
A3 Non-jumping assumption: p11 + p22 > 0 and p33 + p44 > 0
We will now proceed in two steps. First, in Subsection 2.1 we will characterize the evolution of agents’beliefs within a block for arbitrary prior beliefs. Second, in Subsection 2.2 we
will combine these results with di¤erent assumptions regarding agents’prior beliefs following a
switch across blocks. For each of these cases, we will describe how to recast the model with information frictions as a perfect information rational expectations model obtained by expanding
the number of regimes to keep track of agents’beliefs.
2.1
Evolution of Beliefs Within a Block
In what follows, we will derive the law of motion of agents’ beliefs conditional on being in a
speci…c block. The formulas derived below will provide a recursive law of motion for agents’
beliefs based on Bayes’ theorem. Such recursion applies for any starting values for agents’
beliefs. These will be determined by agents’beliefs at the moment the system enters the new
block. We will characterize these initial beliefs in the next subsection.
5
As we have noticed in the previous section, agents can infer the history of the blocks.
Therefore, at each point in time, agents know the number of consecutive periods spent in the
current block since the last switch. Let us denote the number of consecutive realizations of
Block i at time t as it , i 2 f1; 2g. To …x ideas, suppose that the system is in Block 1 at time
t, implying that 1t > 0 and 2t = 0. Then, there are only two possible outcomes for the next
period. The economy can spend an additional period in Block 1, implying that 1t+1 = 1t + 1
and 2t+1 = 0; or it can move to Block 2, implying 1t+1 = 0 and 2t+1 = 1: In this subsection,
we restrict our attention to the …rst case.
Using Bayes’theorem and the fact that prob t 1 = 2j 1t 1 = 1 prob t 1 = 1j 1t 1 ; the
probability of being in Regime 1 given that we have observed 1t consecutive realizations of
Block 1, prob ( t = 1j 1t ) ; is given by:1
prob
t
= 1j
1
t
=
prob
t
prob
1 = 1j
1
p21 ) + p21
t 1 = 1j t 1 (p11
1
p21 p22 ) + p21
t 1 (p11 + p12
(3)
+ p22
where 1t = 1t 1 + 1 and for 1t > 1: Notice that for 1t = 1; prob ( t = 1j 1t ) denotes the initial
beliefs that will be discussed in Subsection 2.2. Equation (3) is a rational …rst-order di¤erence
equation that allows us to recursively characterize the evolution of agents’beliefs about being
in Regime 1 while the system is in Block 1. The probability of being in Regime 3 given that
we have observed 2t consecutive realizations of Block 2, prob ( t = 3j 2t ) ; can be analogously
derived:
prob
t
= 3j
2
t
=
prob
t
prob
1 = 3j
2
p43 ) + p43
t 1 = 3j t 1 (p33
2
p43 p44 ) + p43
t 1 (p33 + p34
+ p44
:
(4)
where 2t = 2t 1 + 1 and for 2t > 1:
The recursive equations (3) and (4) characterize the dynamics of agents’ beliefs in both
blocks for a given set of prior beliefs. In what follows, we will show under which conditions
these beliefs converge. This result will be key to being able to apply the following proposition
and recast Model (1)-(2) in terms of a …nite dimensional set of regimes indexed with respect to
agents’beliefs.
Proposition 1 For any " > 0 there exists a
1
2 N and
2
2 N such that:
prob (
t
= 1j 1 )
prob (
t
= 1j
1
+ 1) < "
prob (
t
= 2j 2 )
prob (
t
= 2j
2
+ 1) < "
Proof. Easy to see once it has been proved that under the required conditions the rational
1
A detailed derivation of equation (3) is provided in Appendix A.
6
di¤erence equations (3) and (4) converge. The propositions below prove this point in steps.
We will characterize the convergence of prob ( t = 1j 1t ) as the number of consecutive periods
spent in Block 1 1t grows large. We will denote 1lim prob ( t = 1j 1t ) = x using prob ( t = 1j 1t ) !
t !1
x and the characteristic roots of equation (3) with :
e1
e2
q
(p11
2p21
p22 )2 + 4p21 p12
p11
p22
p11
2 (p11 + p12 p21 p22 )
q
p22 2p21 + (p11 p22 )2 + 4p21 p12
2 (p11 + p12
p21
(5)
(6)
p22 )
The following propositions provide the conditions under which the di¤erence equation (3)
converges to the stable root e2 . An analogous pair of roots, e3 and e4 , with e4 being the stable
root, can be derived for Block 2. Similarly, all results that follow will also apply to Block 2.
Proposition 2 If (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 6= p22 or both
p12 6= 0 and p21 6= 0, and the initial probability is such that prob ( t = 1j 1t = 1) 6= e1 ; then
prob ( t = 1j 1t ) ! e2 2 [0; 1]. If conditions (i), (ii), and (iii) hold and the initial probability
is such that prob ( t = 1j 1t = 1) = e1 , then prob ( t = 1j 1t ) = e1 for any 1t .
Proof. See Appendix B.
The next proposition relaxes condition (iii) of the above proposition.
Proposition 3 If (i) p11 + p12
p12 = 0 or p21 = 0, then prob (
p21 = 0) or one (if p12 = 0).
t
p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 = p22 and either
= 1j 1t ) ! e1 = e2 and the roots are either equal to zero (if
Proof. See Appendix C.
If the two regimes have the same persistence (p11 = p22 ) and the system has remained in
Block 1 for su¢ ciently long, then agents will eventually believe they are in the regime that is an
absorbing state (conditional on staying in the block). The next proposition relaxes condition
(ii) of the previous propositions.
Proposition 4 If (i) p11 + p12
p11 p21
:
p11 +p12 p21 p22
p21
p22 6= 0, (ii) p11 p22 = p21 p12 , then prob (
t
= 1j 1t ) =
Proof. See Appendix D.
Note that if conditions (i) and (ii) are satis…ed, prob ( t = 1j 1t ) suddenly converges by
jumping to (p11 p21 ) = (p11 + p12 p21 p22 ) as the system enters Block 1. The recursion
(3) can be shown to become a linear di¤erence equation. The solution of this equation is
characterized in the following two propositions.
7
Proposition 5 If (i) p11 + p12 p21
p21
, with p22 pp1121+2p21 2 [0; 1] :
p22 p11 +2p21
p22 = 0 and (ii) p11 6= p21 , then prob (
t
= 1j 1t ) !
Proof. See Appendix E.
Proposition 6 If (i) p11 + p12
p21
p22 = 0, (ii) p11 = p21 , then prob (
t
= 1j 1t ) =
p21
:
p22 +p21
Proof. See Appendix F.
21
When p11 = p21 , beliefs prob ( t = 1j 1t ) suddenly jump to p22p+p
for any 1t
1 (as the
21
system enters Block 1).
To sum up, given the benchmark assumptions A1-A3, we have shown that equation (3)
always converges. Note that Proposition 2 implies that beliefs do not converge to e2 , if the
starting beliefs prob ( t = 1j 1t = 1) = e1 . It can be proved that e1
0 or e1
1, implying
2
that the only admissible values for probabilities are either zero or one. Therefore, there are
only a few limiting cases in which equation (3) does not converge to e2 . As will become clear
later on, it is su¢ cient to set the probability ratios 0 < pi3 = (pi3 + pi4 ) < 1 for any i 2 f1; 2g to
rule out these cases that are not very relevant in practice.
2.2
Evolution of Beliefs Across Blocks
In the previous subsection, we characterized the evolution of agents’ beliefs conditional on
being in a speci…c block. The formulas derived above apply to any set of initial beliefs. In this
subsection, we will pin down agents’beliefs at the moment the economy moves across blocks.
These beliefs will serve as starting points for the recursions (3) and (4) governing the evolution
of beliefs within a block.
Suppose for a moment that before switching to the new block, agents could observe the
regime that was in place in the old block. Notice that in this case the transition matrix conveys
all the information necessary to pin down agents’prior beliefs about the regime in place within
the new block. Speci…cally, we have that if the economy moves from Block 2 to Block 1, the
probability of being in Regime 1 is given by
prob
t
= 1j
t 1
= 3;
1
t
=1 =
p31
;
p31 + p32
if the economy was under Regime 3 in the previous period, or by
prob
2
t
= 1j
t 1
= 4;
A proof is provided in Appendix G.
8
1
t
=1 =
p41
p41 + p42
if the economy was under Regime 4 in the previous period. Symmetrically, the probability of
being in Regime 3 given that the economy just moved to Block 2 is given by
prob
t
= 3j
t 1
= 1;
2
t
=1 =
p13
;
p13 + p14
if the economy was under Regime 1 in the previous period, or by
prob
t
= 3j
t 1
= 2;
2
t
=1 =
p23
p23 + p24
if the economy was previously under Regime 2.
However, in the model, agents never observe the regime that is in place. Therefore, their
beliefs at the moment the economy moves from one block to the other will be a weighted average
of the probabilities outlined above. The weights, in turn, will depend on agents’beliefs at the
moment of the switch. In what follows we will focus on three cases:
1. Static prior beliefs. In this case, the transition matrix P is such that every time the
economy enters a new block, agents’beliefs about which regime has been realized do not
depend on their beliefs right before the switch. Thus, what has been observed in the past
block does not help agents to form expectations in the new block. This restriction has
the virtue of delivering a nice closed-form analytical characterization for the dynamics of
beliefs.
2. Dynamic prior beliefs. In this case, the transition matrix P is such that beliefs about
which regime is prevailing within a block a¤ect prior beliefs the moment the economy
moves to the new block.
3. Signals. Exogenous signals $t about the current regime are also observed by agents.
Signals are assumed to be distributed according to p ($t j t ).
It is worth clarifying that nothing prevents the researcher from combining the three cases
described above. For example, static prior beliefs could characterize one block but not another
or agents could receive a signal every time the economy enters a new block.
2.2.1
The Case of Static Prior Beliefs
In the case of static prior beliefs, every time the system enters a new block, agents’beliefs are
the same regardless of the history of past beliefs. It is immediate to show that necessary and
9
su¢ cient conditions for this to happen are:
prob
t
= 1j
1
t
=1
prob
t
= 3j
2
t
=1
p41
p31
=
= prob
p31 + p32
p41 + p42
p13
p23
=
=
= prob
p13 + p14
p23 + p24
=
t
= 1j
1
t
=1
(7)
t
= 3j
2
t
=1
(8)
In other words, when the economy leaves a block, the relative probability of the two regimes in
the new block is not a¤ected by the regime that was in place before. Agents are fully rational
and know the transition matrix governing the evolution of regimes. Therefore, their beliefs are
uniquely pinned down by (7) and (8).
The recursive equations (3) and (4) combined with the initial conditions (7) and (8) uniquely
characterize the dynamics of agents’beliefs in each block. To see this, notice that for each block,
there is a unique path for the evolution of agents’beliefs, given that (7) and (8) make agents’
beliefs before entering the block irrelevant. Furthermore, Proposition 1 guarantees that there
exists a 1 2 N and 2 2 N such that agents’beliefs converge for an arbitrary level of accuracy.
Therefore, in the case of static priors the number of consecutive periods spent in a block ( it )
is a su¢ cient statistic to pin down the dynamics of beliefs in both blocks. Equipped with this
important result, we can re-cast Model (1)-(2) in terms of a new set of regimes indexed with
respect to the number of consecutive periods spent in a block it , i 2 f1; 2g:
0
( t ) St =
c
( t) +
1
( t ) St
1
+
( t ) "t +
(9)
t
where "t
N (0; " ) is a vector of exogenous Gaussian shocks, t is a vector of endogenous
expectation errors, and the 1 + 2 regimes t
( 1t ; 2t ) evolve according to the transition
matrix
"
#
e11 P
e12
P
e=
P
;
e21 P
e22
P
e11 and P
e12 are given by
where the matrices P
2
e11
P
1
t+1
0 prob
0
..
.
6
6
6
6
6
6
6 0
4
0
= 2j
0
..
.
1
t
=1
::: 0
::: 0
..
. 0
: : : 0 prob f
: : : 0 prob
0
0
e12
P
2
6
6
4
1
prob
1
t+1
= 2j
..
.
1
prob
1
t+1
>
10
j
1
t
=1
1
t
=
3
0
0
0
1
j 1t =
t =
1
j 1t =
t+1 >
01
(
1)
..
.
01
(
1)
3
7
7
5
7
7
7
7
7
7
1g 7
5
with the elements of the matrices given by
prob
i
t+1
=
i
t
+ 1j
i
t
= prob
t
= 1j
1
t
(p11 + p12 ) + 1
prob
t
= 1j
1
t
(p21 + p22 ) (10)
where prob ( t = 1j 1t ) can be obtained from the recursive equation (3) and equation (7). The
e21 and P
e22 can be analogously derived.
matrices P
Notice that the newly de…ned set of regimes keeps track of both the parameters in place
at each point in time and the evolution of agents’ beliefs. Since Model (9) is a Markovswitching DSGE model with perfect information, it can be solved using the techniques developed
by Schorfheide (2005), Liu, Waggoner, and Zha (2011), Davig and Leeper (2007), Farmer,
Waggoner, and Zha (2009), Cho (2012), and Foerster, Rubio-Ramirez, Waggoner, and Zha
(2011). The result is an MS-VAR in the DSGE state vector St :
St = c
e +T
t; P
e St
t; P
1
+R
e "t
t; P
(11)
where the law of motion of the economy depends on agents’beliefs as captured by t . With
the results of Proposition 1 at hand, the solution of Model (9) with a truncated number of
regimes t approximates the solution of the original model with learning (1). Notice that
the accuracy of this approximation can be made arbitrarily precise simply by increasing the
number of regimes . Furthermore, it is worth pointing out that in the case of static priors
the approximation error stems only from truncating agents’learning process. For all regimes
such that it < i agents’beliefs exactly coincide with the analytical values derived using (3)
and (4) and conditions (7) and (8).
2.2.2
The Case of Dynamic Prior Beliefs
When conditions (7) and (8) do not hold, past beliefs always in‡uence current beliefs. In this
case, the number of consecutive periods t spent in a block is no longer a su¢ cient statistic for
agents’beliefs. However, as pointed out before, the recursive equations (3) and (4) hold for any
prior beliefs. Therefore, these equations still capture the dynamics of beliefs while the system
stays in a block. Furthermore, it follows that the su¢ cient conditions for convergence derived
in Subsection 2.1 still apply. Nevertheless, the initial conditions are now di¤erent from (7) and
(8) as they will depend on beliefs in the past block. Speci…cally, agents’starting beliefs upon
the shift from Block 2 to Block 1 are given by
prob f
t
= 1jIt g =
prob
prob t 1 = 3jIt 1 p31 + 1
t 1 = 3jIt 1 (p31 + p32 ) + 1
11
prob
prob
t 1
t 1
= 3jIt
= 3jIt
1
1
p41
(p41 + p42 )
(12)
while if the system just entered Block 2, starting beliefs read
p23
prob
(p23 + p24 )
1
t 1
(13)
Notice that, using their information set It ; agents can keep track of both the number of
consecutive deviations and their starting beliefs. Therefore, in the case of dynamic prior beliefs
two variables pin down the dynamics of beliefs over time: how many consecutive periods the
system has spent in the current block and the initial beliefs agents had when the system
entered the current block. We then tackle the problem of solving Model (1) when prior beliefs
are dynamic by making a grid for agents’beliefs. Denote the grid for beliefs prob f t = 1jIt g as
Gb1 = fG1 ; :::; Gg1 g and for beliefs prob f t = 3jIt g as Gb2 = fGg1 +1 ; :::; Gg1 +g2 g where 0 Gi 1,
all 1 i g = g1 + g2 . Furthermore, we denote the whole grid as G = G b1 [ Gb2 . Endowed with
such a grid, we can recast the original model in terms of a new set of regimes t 2 f1; :::; g1 +g2 g,
any t. The new regime t captures the knot of the grid G that best approximates agents’beliefs;
that is, in our notation prob f t = 1jIt g when the system is in Block 1 and prob f t = 3jIt g when
the system is in Block 2. The transition probability matrix for these new regimes can be pinned
down using the recursions (3) and (4) and the initial conditions (12) and (13). The algorithm
below illustrates how exactly to perform this task.
prob f
t
prob t 1 = 1jIt 1 p13 + 1
t 1 = 1jIt 1 (p13 + p14 ) + 1
= 3jIt g =
prob
prob
b for the new regimes
Algorithm Initialize the transition matrix P
t 1
t,
= 1jIt
= 1jIt
1
b = 0g g :
setting P
Step 1 For each of the two blocks, do the following steps (without loss of generality we describe
the steps for Block 1):
Step 1.1 For any grid point Gi 2 Gb1 ; 1
b (i; j) = prob
P
t 1
= 1jIt
1
g1 , compute
i
(p11 + p12 ) + 1
prob
t 1
= 1jIt
1
(p21 + p22 )
where prob t 1 = 1jIt 1 = Gi and j g1 is set so as to min jprob f t = 1jIt g Gj j,
where prob f t = 1jIt g is computed using the recursive equation (3) along with the
approximation prob t 1 = 1jIt 1 = Gi . To ensure the convergence of beliefs, we
correct j as follows: if j = i and Gi 6= e2 , then set j = min (j + 1; g1 ) if Gi < e2 or
j = max (1; j 1) if Gi > e2 .
Step 1.2 For any grid point Gi 2 Gb1 ; 1
l > g1 satisfying
min
prob
i
prob t 1 = 1jIt 1 p13 + 1
t 1 = 1jIt 1 (p13 + p14 ) + 1
12
b (i; l) = 1
g1 ; compute P
prob
prob
t 1
t 1
= 1jIt
= 1jIt
1
1
b (i; j) with
P
p23
(p23 + p24 )
Gl
where prob
t 1
= 1jIt
1
= Gi .
b has all zero elements, stop. Otherwise, go to Step 3.
Step 2 If no column of P
P
b (i; j) 6=
Step 3 Construct the matrix T as follows. Set j = 1 and l = 1. While j g, if gi=1 P
0 then do three things: (1) set T (j; l) = 1, (2) set T (j; v) = 0 for any 1
v
g and
Pg b
v 6= l, (3) set l = l + 1 and (4) set j = j + 1; otherwise (i.e., if i=1 P (i; j) = 0), set
j = j + 1.
bR = T P
b T 0 . If no column of P
bR has all zero
Step 4 Write the transition equation as P
b=P
bR and stop. Otherwise, go to step 3.
elements, set P
Step 1.1 determines the regime j the system will go to if it stays in Block 1 next period and
b with the probability of moving
…lls up the appropriate element (i; j) of the transition matrix P
to Regime j. If j = i and the corresponding diagonal element is not the knot that beliefs
converge to, the step makes a correction to prevent this from happening. Otherwise, beliefs
could get stuck at a grid point that is not the convergence point, inaccurately representing the
dynamic of beliefs.3 Step 1.2 computes the regime l the system will go to if it leaves Block 1
b Steps 2-4 are not necessary but help
and …lls up the appropriate element (i; l) of matrix P.
to keep the dimension of the grid small, getting rid of regimes that will never be reached. For
computational convenience, we always add the convergence points for the two blocks (i.e., e2
in the case of Block 1) to the grid G. On many occasions, it is a good idea to make the grid
near the convergence knot very …ne to improve the precision of the approximation.
b for the new set of regimes is characterized, the original Model
Once the transition matrix P
(1) can be recast in terms of the new set of regimes t :
0
( t ) St =
c
( t) +
1
( t ) St
1
+
( t ) "t +
t
(14)
where t 2 f1; :::; g1 +g2 g. Therefore, up to an approximation error that can be made arbitrarily
small, the task of solving the model with learning in (1) boils down to solving the perfectinformation model (14) using solution algorithms for MS-DSGE models. The resulting law of
motion is once again an MS-VAR:
St = c
3
e +T
t; P
e St
t; P
1
+R
e "t
t; P
(15)
Similarly, in the case of oscillatory convergence, one has to prevent the algorithm from getting stuck on
some grid point that does not represent the convergence point. To avoid complicating the discussion of the
paper with too many technical details, we omit this correction.
13
2.2.3
Signals
Let us assume that agents observe signals about the realized regime. To …x notation, denote
the signal as $t and, for simplicity, assume that it can have only two values, 1 or 2. We denote
the probability that the signal is equal to q 2 f1; 2g, conditional on the regime being equal to
h 2 f1; 2; 3; 4g as prob f$t = qj t = hg. The model with signals can be solved by introducing
a new system of regimes t , which indexes the grid points corresponding to the probabilities
prob f t = 1jIt ; $t g and prob f t = 3jIt ; $t g, and following the same logic used in the previous
subsection.
b for the new set of regimes, one can implement the alTo …ll up the transition matrix P
gorithm detailed in Subsection 2.2.2 with only the little tweak of updating beliefs using the
information contained in the observed signal. For instance, we compute the ex-post-probability
prob ( t = 1jIt ; $t )
prob ($t = qj t = 1) prob ( t = 1jIt ; $t 1 )
t 1
; q 2 f1; 2g
=
1jI
;
$
;
$
=
q
=
P
t
t
t
2
t 1)
i=1 prob ($ t = qj t = i) prob ( t = ijIt ; $
(16)
t 1
where prob ( t = 1jIt ; $ ) is computed using the recursive equation (3) for a given initial point
in the grid G that approximates prob t 1 = 1jIt 1 ; $t 1 . We use the probability computed
in equation (16) to determine the appropriate points of the grid G, which we denote as jq ,
q 2 f1; 2g. Note that for any given initial belief prob t 1 = 1jIt 1 ; $t 1 2 G, the (ex-post)
belief prob ( t = 1jIt ; $t 1 ; $t = q) now pins down the grid points, depending on the realization
of the signal $t . Once these two points in the grid are determined, we can …ll up the transition
probability as follows:
prob
where
b (i; jq ) = P2 prob
P
v=1
prob
t
t
= vjIt 1 ; $t
= vjIt 1 ; $t
1
=
P2
u=1
1
prob f$t = qj
prob
t 1
t
= vg ; q 2 f1; 2g
= ujIt 1 ; $t
1
puv
(17)
(18)
and we approximate prob t 1 = 1jIt 1 ; $t 1 2 G. Note that in the case of signals, each row
b has up to four non-zero elements. This completes the derivation of
of the transition matrix P
b11 , which governs the evolution of beliefs within Block 1. How to obtain the
the submatrix P
b12 , P
b21 , and P
b22 is detailed in Appendix H.
other submatrices P
2.3
Agents Know That They Do Not Know
Summarizing, the methods outlined above show that one can recast the Markov-switching
DSGE model with learning as a Markov-switching rational expectations system, in which the
14
regimes are indexed with respect to agents’beliefs. In the case of static priors, the number of
consecutive realizations of a block represents a su¢ cient statistic to index agents’beliefs. In the
case of dynamic priors, agents’beliefs are mapped into a grid. In both cases, a new transition
matrix that characterizes the joint evolution of agents’beliefs and model parameters is derived.
It is worth emphasizing that this way of recasting the learning process allows us to easily
model economies in which agents know that they do not know. In other words, agents form
expectations taking into account that their beliefs will change in the future according to what
they will observe in the economy. This is why the laws of motion (11) and (15) characterizing the
behavior of the model depend on the current beliefs and the expanded transition matrix de…ning
the joint evolution of agents’ beliefs and model parameters. This represents a substantial
di¤erence with the anticipated utility approach in which agents form expectations without
taking into account that their beliefs about the economy will change over time (e.g., Evans
and Honkapohja, 2001; Cogley, Matthes, and Sbordone, 2011). Furthermore, the approach
described above di¤ers from the one traditionally used in the learning literature in which agents
form expectations according to a reduced-form law of motion that is updated recursively using
the discounted least-squares estimator (Eusepi and Preston, 2011). The advantage of adaptive
learning is the extreme ‡exibility given that, at least in principle, no restrictions need to be
imposed on the type of parameter instability characterizing the model. However, such ‡exibility
does not come without a cost, given that agents are not really aware of the model they live
in, but only of the implied law of motion. Instead, in this paper agents fully understand the
model, they are uncertain about the future, and they are aware of the fact that their beliefs
will evolve over time.
It is also important to emphasize the extreme tractability of the approach taken in this paper.
The solutions (11) and (15) can be easily combined with an observation equation and used in an
estimation algorithm. For example, Bianchi and Melosi (2012a) estimates a prototypical NewKeynesian DSGE model, in which agents form beliefs about the likely duration of deviations
from active in‡ation stabilization policies. The estimation of this new class of models is possible
for three main reasons. First, even if the …nal number of regimes can be extremely high, the
model imposes very speci…c restrictions on the allowed regime paths and on the link between
observable outcomes and agents’ beliefs. This implies that when evaluating the likelihood, a
relatively small number of regime paths has to be taken into account. Second, the statistical
properties of the di¤erent regimes can vary substantially and depend on the probability of
moving across regimes. Therefore, identi…cation of the transition matrix is not only given by the
frequency with which the di¤erent regimes occur, but also by the laws of motion characterizing
the di¤erent regimes. Finally, the number of extra parameters with respect to a model with
perfect information is very low, if not zero, while the resulting dynamics can be substantially
enriched. For example, Bianchi and Melosi (forthcoming) show that a period of …scal distress
15
can lead to a run-up in in‡ation that lasts for decades.
From a computational point of view, there might be a concern about the time required to
solve the model when the …nal number of regimes becomes very large. This turns out not to be
a problem. If regime changes enter in an additive way, a¤ecting only the matrix c , the model
can be solved with standard solution algorithms such as gensys (Sims, 2002) and Blanchard and
Kahn (1980) and the high dimensionality of the transition matrix is not a problem. However,
in many situations we might want to model regime changes that enter in a multiplicative way.
For example, we might want to allow for changes in the Taylor rule parameters. In this case,
the matrices 0 and 1 are also a¤ected and we need to rely on solution methods developed
to solve MS-DSGE models. However, according to our experience based on the use of the
approach proposed by Farmer, Waggoner, and Zha (2009), even in this case a solution can be
obtained in a matter of seconds because the transition matrix governing the evolution of the
regimes is very sparse. Therefore, the methods described in this paper provide a promising tools
for modeling information frictions, animal spirits, and agents’beliefs in a general equilibrium
framework suitable for structural estimation.
3
Applications
In this section, we introduce a prototypical RBC model to illustrate the properties of the
methods detailed above. Central to our discussion will be the evolution of optimism and pessimism and the implications thereof for consumption and saving decisions. The representative
household maximizes the sequence of consumption ct and capital kt :
e0 P1
max E
t=0
ct ;kt
t
ln ct
subject to the resource constraint ct + kt = zt kt 1 + (1
) kt 1 with < 1 and 0 < < 1.
e
Let Et ( ) denote the expectation operator conditional on households’information set at time
t. We assume that total factor productivity (TFP) zt follows an exogenous process, such that
ln zt =
( t ) + ln zt
iid
1
+
z "t
(19)
where "t v N (0; 1) and t denotes a discrete Markov process a¤ecting the drift of TFP. This
process evolves according to the transition probability matrix P. We assume that t can take
four values; that is, t 2 f1; 2; 3; 4g. These values map into values for the TFP drift ( t ) as
follows t 2 f1; 2g =) t ( t ) = H and t 2 f3; 4g =) t ( t ) = L , where L < H . In Block
1, Regimes 1 and 2 di¤er in their likely persistence: p11 < p22 . The same applies to Regimes 3
and 4 in Block 2: p33 < p44 . We call Regimes 1 and 2 high-growth regimes and Regimes 3 and
16
4 low-growth regimes. Households are assumed to observe the history of the model variables
(i.e., kt , ct , and zt ) and that of the TFP shocks (i.e., "t ). Therefore, households can establish
whether the economy is in the high-growth block or in the low-growth block.
1=(1
) e
1=(1
)
We introduce the stationary variables t
ln (zt =zt 1 ), e
ct
ct =zt
, kt
kt =zt
and, following Schorfheide (2005) and Liu, Waggoner, and Zha (2011), we de…ne the steady
state as the stationary equilibrium in which all shocks are shut down, including the regime
shocks to the growth rate of TFP. We then derive a log-linear approximation to the equilibrium
equations around the steady-state equilibrium for these stationary variables. The log-linearized
Euler equation reads:4
etb
b
ct = E
ct+1
1
(
1) 1 + (
1
1) M
1
b
kt
1
1
+ M
1
(
et bt+1
1) + 1 E
(20)
(p1 + p2 ) H + (p3 + p4 ) L being the ergodic mean of the log growth
where M exp ( ),
ct and
rate of the economy, and pi stands for the ergodic probability of being in regime i, bt ; b
b
kt denote log-deviations of the stationary TFP growth, consumption, and capital, respectively,
from their steady-state value, and b ( t )
is the log-deviation of TFP drift from its
t ( t)
ergodic mean . The resource constraint is
cssb
ct + kss b
kt =
M
1
kss
1
+
1
1
1
M
1
kss bt + M
1
kss + (1
)
Finally, the log-deviation of the growth rate of TFP from its ergodic level follows
bt = b ( t ) +
z "t
1
1
kss b
kt
1
(21)
(22)
As is standard for any RBC model, households adjust capital so as to smooth consumption
intertemporally. The occurrence of TFP shocks and the succession of low-growth and highgrowth regimes challenge households’ ability to smooth consumption over time. When the
economy is in the high-growth regime, households expect that, with some probability, the
economy will enter into the low-growth regime in the future, making it harder to raise future
consumption. Therefore, ceteris paribus agents raise capital today so as to raise future expected
etb
consumption E
ct+1 vis-a-vis current consumption b
ct . When the economy is in the low-growth
regime, agents expect that, with some probability, the economy will enter into the high-growth
regime in the future, making it easier to raise future consumption. Therefore, ceteris paribus
agents reduce capital today so as to raise current consumption b
ct vis-a-vis expected future
etb
consumption E
ct+1 .
4
A detailed derivation of the steady-state equilibrium for the stationary variables and the log-linearized
equations is provided in Appendix I.
17
Clearly, the persistence of the regime in place critically a¤ects consumption and capital
decisions. When the current regime is short lasting, households generally adjust capital more
aggressively than when it is long lasting, because they deem that a switch in the next period
is more likely. In contrast, households do not adjust capital so aggressively if they expect
the regime to be very long lasting. When households expect that low growth or high growth
has become a structural characteristic of the environment, they understand that consumption
cannot be smoothed out over time by simply adjusting capital. Thus, very persistent regimes
are mostly characterized by structural changes in the level of consumption.
Given that households have limited information, the log-linearized Model (20)-(22) cannot
be solved using the existing techniques that are used to solve Markov-switching models with
perfect information. However, we proceed as described in the previous sections, by introducing
a new set of regimes that capture the evolution of the representative household’s beliefs over
time. It is important to notice that in the RBC model described above, regime changes enter
additively. In other words, they only a¤ect the vector of constants c ( ) in the canonical forms
(9) or (14). In this case, the state space can be augmented with a series of dummy variables as in
Schorfheide (2005), Liu, Waggoner, and Zha (2011), and Bianchi, Ilut, and Schneider (2012) and
the models under imperfect information can be easily solved using standard solution methods
for DSGE model such as gensys (Sims, 2002) and Blanchard and Kahn (1980). When regime
changes enter multiplicatively, the matrices 0 and 1 are also a¤ected. In this case, the model
can be solved with any of the solution methods that have been developed for MS-DSGE models.
Bianchi and Melosi (2012a, 2012b, Forthcoming) consider these cases and solve the model using
the algorithm developed by Farmer, Waggoner, and Zha (2009).
In what follows, we adopt a standard calibration of the RBC model. We set capital’s share
parameter to equal 0:33. The discount factor is equal to 0:9976 and the parameter for
the physical depreciation of capital is set to equal 0:0250. The standard deviation of the TFP
shock is set to 0:007. We set the growth rate of TFP in the high-growth state to equal the
annualized rate of 4%: H = :01. We assume that under low-growth, the growth rate of TFP
is simply zero: L = 0. The values for the transition matrix P will vary across simulations and
will be described case by case.
3.1
Static Prior with Symmetric Speed of Learning
Let us assume that te persistence of the short-lasting regimes is the same in the two blocks:
p11 = p33 = 0:5. Analogously, we set the probabilities of staying in the long-lasting regimes
so that p22 = p44 = 0:95. For simplicity we assume that regimes belonging to the same block
do not communicate with each other; that is, p12 = p21 = p34 = p43 = 0. Furthermore, the
transition matrix implies that once a switch to a new block occurs, agents always attach a 95%
18
Figure 1: Static Prior Beliefs; Expected Growth Rate of Technology as a Function of Beliefs. The scale of the
expected growth rates ranges from -1.6 to 1.6. Lighter blue areas capture expected rates that are lower than
the ergodic rate. Darker red areas capture expected rates that are higher than the ergodic rate. Gray areas
denote expected rates that are similar to the ergodic rate. Left graph: Annualized percentage deviations of the
growth rate of technology from its ergodic value at di¤erent horizons as a function of the probability that the
observed high-growth regime is long lasting (Beliefs LL-HG). Right graph: Annualized percentage deviations of
the growth rate of technology from its ergodic value at di¤ferent horizons as a function of the probability that
the observed low-growth regime is long lasting (Beliefs LL-LG).
probability to being in the short-lasting regime:
p41
p31
=
= 0:95
p31 + p32
p41 + p42
p23
p13
=
= 0:95
p13 + p14
p23 + p24
(23)
(24)
Notice that conditions (23)-(24) imply static prior beliefs: agents always enter the high-growth
(low-growth) block with the same level of optimism (pessimism). To sum up, we work with the
following transition matrix:
2
0:50
6
6
0
P =6
6 0:475
4
0:0475
0
0:95
0:475
0:0475
0:025
0:0025
0:50
0
3
0:025
7
0:0025 7
7
7
0
5
0:95
Figure 1 shows agents’ expectations about the growth rate of TFP t in deviations from
its ergodic level at di¤erent horizon as agents’beliefs about being in the high-growth long-
19
lasting regime (left plot) or in the long-lasting low-growth regime (right plot) vary from 0 to 1.
These expectations take into account regime uncertainty and can be quickly computed using
the methods described in Bianchi (2012). When the economy is in the high-growth block and
agents are extremely optimistic about being in the long-lasting regime, they expect a growth
rate of TFP that exceeds the ergodic rate by 1.6% for the next few quarters. left graph. See the
dark red area in left graph. However, at very long horizons, agents simply expect the growth
rate of TFP to return to the annualized ergodic mean of 2%, regardless of how con…dent they
are to be in the long-lasting high-growth regime. When agents mostly expect to be in a shortlasting regime, the expected growth rate of TFP is fairly similar to the ergodic rate at every
horizon. Similarly, when the economy is on a low-growth path, the more pessimistic about the
duration of this phase agents are, the longer the horizon at which the expected growth rate of
TFP stays below the ergodic rate. See the light blue area in the right graph.
To illustrate the consequences of ‡uctuations in agents’ beliefs, we simulate the economy
assuming a typical path for the regimes and setting all Gaussian shocks "t to zero. We assume
that capital is initialized at its steady-state value. The results are reported in Figure 2. In
each panel, the gray areas denote the periods of low growth. Short-lasting regimes last for
their typical duration of 2 quarters. Long-lasting regimes last for their typical duration of 20
quarters. The two right graphs report the evolution of consumption and capital in the model
with learning compared to the model with perfect information in which agents can observe
the current regime. The panel in the upper-left corner shows the evolution of agents’ beliefs
about being in the long-lasting high-growth regime and in the long-lasting low-growth regime.
The panel in the lower-left corner reports the evolution of expected average TFP growth at
4-, 8-, 20-, and 40- quarter horizons. Notice that this is a convenient measure of agents’
optimism/pessimism that takes into account uncertainty about the regime in place today and
the possibility of regime changes.
Three features of Figure 2 deserve to be emphasized. First, right after a switch to a new
block, agents believe that this switch is most likely to be short lasting. This can be seen in the
top left graph when switches to new blocks occur. The reason is that agents are rational and
hence are aware that regardless of whether the past regime was short lasting or long lasting, the
probability of switching to the short-lasting regime in the new block is always as high as 95%.
This stems from the restrictions in (23)-(24), which imply static prior beliefs. Second, whenever
a short-lasting regime is in fact realized, with the bene…t of hindsight, agents’beliefs turn out
to over-react to the regime change because agents rationally attach a non-zero probability to
being in the long-lasting regime. Third, the probability of being in the long-lasting regime
smoothly increases as more realizations of the same block are observed. The top left graph
shows that the probability of being in the long-lasting regime rises monotonically with the
number of consecutive realizations of a particular growth rate. For instance, from t = 117 to
20
Pr. LL-HG
Pr. LL-LG
0.8
Beliefs
Consumption Gap
1
0.4
0.2
0.6
50
100
150
200
5
4
3
2
1
0
50
100
150
200
50
100
150
200
5
4Q
8Q
20Q
40Q
Ergodic
Capital Gap
E(Avg TFP Growth)
1
15
10
5
0
-5
-10
-15
50
100
150
3
1
-1
-3
-5
200
Figure 2: Static Prior Beliefs and Symmetric Speed of Learning. Top left graph: Evolution of beliefs of being in
the long-lasting high-growth regime (red solid line) and in the long-lasting low-growth regime (blue dashed line).
Top right graph: Annualized percentage deviations of consumption from the perfect-information benchmark.
Bottom left graph: Expected average growth rate of technology (annualized percentage) at various horizons.
Bottom right graph: Annualized percentage deviations of capital from the perfect-information benchmark. In
all graphs, gray areas denote periods of low growth.
t = 136, the economy is in a long-lasting low-growth regime. While agents initially attach
a small probability to being in the long-lasting regime, they become fully convinced after 12
consecutive periods of low TFP growth.
Furthermore, Figure 2 shows the evolution of optimism and pessimism and the associated
dynamics of the consumption gap and the capital gap, which are de…ned as the optimal level of
consumption and capital in deviations from their corresponding levels under perfect information.
When the economy enters the long-lasting low-growth period, imperfectly informed agents are
not very pessimistic about the duration of the low-growth regime. This is re‡ected in their
expectations about the average growth rate of TFP that barely moves in the bottom left graph.
Given that they expect that the low-growth period will be short lasting, they decide to slow
down capital accumulation so as to smooth consumption. In contrast, if agents knew the actual
realization of the low-growth regime, they would adjust their stock of capital less aggressively
and let consumption fall more. This is why in Figure 2 we observe a positive consumption gap
and a negative capital gap when the economy enters a period of long-lasting low growth.
As the period of low-growth consolidates, imperfectly informed agents update their beliefs
until they eventually become convinced that they are in the long-lasting regime. This happens
in roughly 12 quarters after the switch. As illustrated in the bottom left graph, such slow21
moving beliefs cause the expected average growth rate of TFP over the next few years to also
adjust sluggishly. This eventually determines an adjustment in the path for consumption and
the consumption gap slowly fades away. Interestingly, at the end of the long-lasting low-growth
period, the consumption gap becomes negative. The reason is that the sluggish evolution of
pessimism prompted households to decumulate capital rapidly at the beginning of the period of
low growth. The small capital stock depresses consumption as households become pessimistic,
leading to a negative consumption gap. A specular pattern characterizes the economy the
moment it enters a long-lasting high-growth period at the beginning of the simulation.
As pointed out before, even when the economy repeatedly alternates between short-lasting
periods, agents’beliefs are misaligned with the truth. Let us focus on the …rst 16 quarters during
which a sequence of short-lasting regimes are realized. While the economy is in the short-lasting
high-growth regime, imperfectly informed households consume more and accumulate less capital
than in the case of perfect information. The reason is that imperfectly informed agents attach
some non-negligible - albeit small - probability to being in the long-lasting regime. By the same
token, when the economy is going through a short-lasting period of low growth, imperfectly
informed households consume less and accumulate more capital than under perfect information.
3.2
Static Prior with Di¤erential Speed of Learning
Now we study a situation in which optimism and pessimism evolve at di¤erential speed. The
speed of learning within a block is a¤ected by the relative persistence of the corresponding two
regimes. Generally speaking, if the persistences of the two regimes become more similar, it
takes longer for rational agents to …gure out which regime is in place. To see this point, let us
calibrate the model as in the previous section, using the restrictions in (23)-(24), but now we set
the probability of staying in the short-lasting high-growth regime to be p11 = 0:75 > 0:5. The
probability of staying in the long-lasting high-growth regime is unchanged (p22 = 0:95). Figure
3 shows the dynamic of beliefs, the expected average growth rate of TFP at various horizons
(4, 8, 20, and 40 quarters), the consumption gap, and the capital gap when the economy goes
through the same sequence of regimes as that in Figure 2, with the only di¤erence that now the
typical duration of the short-lasting high-growth regime is longer: 4 quarters instead of 2. The
typical duration of all other regimes is the same. The crucial point to notice is that in Figure
3 the typical realization of 20 quarters of high growth is not enough to induce households fully
make up their mind that the realized regime is of the long-lasting type. Agents attach only an
80% probability of being in the long-lasting regime after having observed 20 consecutive periods
of high growth. In contrast, when the economy is going through a period of long-lasting low
growth, it takes roughly 12 quarters for households to be fully convinced that they are in the
long-lasting regime, exactly as in Figure 2.
22
Pr. LL-HG
Pr. LL-LG
0.8
Beliefs
Consumption Gap
1
0.4
0.2
0.6
50
100
150
200
250
300
5
5
4
3
Capital Gap
E(Avg TFP Growth)
1
15
10
5
0
-5
-10
-15
3
2
4Q
8Q
20Q
40Q
Ergodic
1
0
50
100
150
200
250
100
150
200
250
300
50
100
150
200
250
300
1
-1
-3
-5
300
50
Figure 3: Static Prior Beliefs and Di¤erential Speed of Learning. Top left graph: Evolution of beliefs of being in
the long-lasting high-growth regime (red solid line) and in the long-lasting low-growth regime (blue dashed line).
Top right graph: Annualized percentage deviations of consumption from the perfect-information benchmark.
Bottom left graph: Expected average growth rate of technology (annualized percentage) at various horizons.
Bottom right graph: Annualized percentage deviations of capital from the perfect-information benchmark. In
all graphs, gray areas denote periods of low growth.
Di¤erential speeds of learning have an impact on the dynamics of consumption and capital. During the long-lasting high-growth regime, the misalignment of agents’ beliefs is more
persistent than in the case of the low-growth regime. This implies a more persistent negative
consumption gap because households raise capital to smooth consumption in the future. On
the other hand, the consumption gap is less pronounced than that under a symmetric speed of
learning. The reason is that the expected duration of the short-lasting high-growth regime is
now longer than that in the previous subsection. This is captured by the graph in the lower
left corner that shows a larger jump in the expected average growth rate as the economy enters
the high-growth block when compared to its counterpart in 2.
3.3
Evolution of Optimism and Pessimism
So far, we have assumed that prior beliefs are static. Static prior beliefs imply that previous
beliefs do not a¤ect current beliefs once the observed TFP growth rate changes. Let’s consider
a transition matrix of the following type. We assume that the probability of staying in the
short-lasting regimes is p11 = p33 = 0:75. We set the probabilities of staying in the long-lasting
regimes so that p22 = p44 = 0:95. For simplicity we assume that the regimes belonging to the
23
Figure 4: Dynamic Prior Beliefs; Expected Growth Rate of Technology as a Function of Beliefs. The scale of
the expected growth rates ranges from -1.6 to 1.6. Lighter blue areas capture expected rates that are lower than
the ergodic rate. Darker red areas capture expected rates that are higher than the ergodic rate. Gray areas
denote expected rates that are similar to the ergodic rate. Left graph: Annualized percentage deviations of the
growth rate of technology from its ergodic value at di¤erent horizons as a function of the probability that the
observed high-growth regime is long lasting (Beliefs LL-HG). Right graph: Annualized percentage deviations of
the growth rate of technology from its ergodic value at di¤ferent horizons as a function of the probability that
the observed low-growth regime is long lasting (Beliefs LL-LG).
same block do not communicate with each other: p12 = p21 = p34 = p43 = 0. We will relax this
restriction later on.
We want to model an economy that goes through two types of phases over time: a highgrowth phase that is mostly characterized by long-lasting high-growth periods with only very
short low-growth periods and a low-growth phase that is mostly characterized by persistent
periods of low-growth and high-growth periods of rather short duration. This can be done by
introducing the following restrictions on the parameters of the transition matrix P:
p41
p31
= 0:05 <
= 0:95
p31 + p32
p41 + p42
p23
p13
= 0:05 <
= 0:95
p13 + p14
p23 + p24
24
(25)
(26)
To sum up, the transition matrix reads:
2
0:75
6
6
0
P =6
6 0:0125
4
0:0475
0
0:95
0:0125
0:0475
0:2375
0:0025
0:75
0
3
0:2375
7
0:0025 7
7
7
0
5
0:95
It is important to emphasize that, in this model, the fact that the economy is currently
in the high-growth or low-growth regime plays a minor role in a¤ecting the growth rate that
agents expect. Most of the action stems from whether agents believe that the economy has been
going through a high-growth phase or a low-growth phase. Figure 4 shows agents’expectations
about the growth rate of TFP t in deviations from its ergodic level at di¤erent horizons
(from 1 to 80 quarters) and for various initial levels of probability of being in the long-lasting
high-growth regime (left plot) and low-growth regime (right plot). Notice that, unlike in the
case of static prior beliefs discussed in Figure 1, high-growth regimes may be associated with a
lower medium-horizon expected growth rate of TFP than low-growth regimes. This is because,
in this economy, agents know that short-lasting regimes are more likely to be followed by the
long-lasting regime of the opposing block. Importantly, the expected growth rate of technology
in the long-lasting high-growth (low-growth) regime di¤ers from that in the short-lasting lowgrowth (high-growth) regime only at a very short horizon.
The upper left graph of Figure 5 reports the evolution of agents’beliefs, consumption, and
capital for the case of dynamic prior beliefs. We assume a typical path for the regimes. We
initialize agents’ beliefs so that agents are con…dent of being in a high-growth phase.5 As
agents observe 4 quarters of high growth, followed by 20 quarters of low growth, agents start to
fear that the economy has switched to the low-growth phase. As a result, households are less
optimistic when the economy returns to the high-growth regime. When the second realization
of the long-lasting low-growth regime occurs, households become immediately convinced that
the long-lasting low-growth regime is in place. Symmetrically, when the economy returns to
high TFP growth for the second time, households believe that the high-growth regime will be
long-lasting with only a 6% probability. Afterwards, the economy enters the high-growth phase
by going through a short-lasting low-growth regime. Households are initially very pessimistic
about the persistence of this regime. It takes two realizations of the long-lasting high-growth
regimes to make them con…dent that the economy has shifted to the high-growth phase.
The lower left graph of Figure 5 provides further evidence that households slowly learn
about changes in the two paths. Observe that when the economy enters the …rst long-lasting
low-growth period, households mostly believe that they are still in the high-growth phase and
5
This can easily happen if the economy just went through a high-growth phase.
25
Consumption Gap
1
Beliefs
0.8
0.4
0.2
0.6
25
50
75
100
5
4
3
2
1
0
10
-5
-20
-35
25
50
75
100
25
50
75
100
20
4Q
8Q
20Q
40Q
Ergodi c
Capital Gap
E(Avg TFP Growth)
1
Pr. LL-HG
Pr. LL-LG
25
25
50
75
10
0
-10
-20
100
Figure 5: Dynamic Prior Beliefs; Expected Growt Rate of Technology as a Function of Beliefs. Left graph:
Annualized percentage deviations of the growth rate of technology from its ergodic value at di¤erent horizons as
a function of the probability that the observed high-growth regime is long lasting (Beliefs LL-HG). Right graph:
Annualized percentage deviations of the growth rate of technology from its ergodic value at di¤ferent horizons
as a function of the probability that the observed low-growth regime is long lasting (Beliefs LL-LG).
expect an average growth rate of TFP over the next 20 or 40 quarters that is above the ergodic
level. The same sluggishness in the expected average growth rate of TFP can be observed
as the economy enters the …rst long-lasting high-growth period. Furthermore, the sluggish
dynamics of optimism and pessimism are con…rmed by a quick comparison of the expected
average growth rate of TFP across short-lasting periods. It is important to notice that the
earliest short-lasting high-growth (low-growth) regime is associated with rates that are largely
below (above) the ergodic rate at all horizons.
The behavior of consumption and capital during the low-growth and the high-growth phase
is analyzed in the rigthside graphs of Figure 5. We observe that at the beginning of the …rst
short-lasting high-growth regime, which is associated with high optimism, the consumption gap
is positive. The reason is that imperfectly informed households expect this regime to be much
longer lasting than what it actually turns out to be. This implies that imperfectly informed
households do not raise capital as they would if they knew that the high-growth regime is, in
fact, short lasting. This leads to a negative capital gap and a positive consumption gap. When
the economy enters the long-lasting low-growth regime for the …rst time, the consumption gap
remains positive as households are not very pessimistic about the persistence of this regime.
This happens because households decide to cut capital fairly aggressively in order to sustain
current consumption, since they mostly expect this low-growth regime to be short lasting.
26
Prob LL-HG Regime
Prob LL-LG Regime
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
1
True Beliefs
Approximated Beliefs
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
Figure 6: Accuracy. Top graph: True and estimated path of agents’ beliefs of being in the long-lasting
high-growth regime. Lines with squares: approximated beliefs. Lines with circles: true beliefs. Bottom graph:
True and estimated path of agents’beliefs of being in the long-lasting low-growth regime. Lines with squares:
approximated beliefs. Lines with circles: true beliefs.
Households would do otherwise, if they knew that the economy just entered the long-lasting
low-growth regime.
During the …rst long-lasting low-growth spell households update their beliefs until they
realize that this regime is most likely long lasting, signifying that the economy must have
switched to the low-growth phase. This change in agents’beliefs causes consumption and capital
(the latter with some sluggishness) to become similar to the perfect-information benchmark.
Interestingly, the consumption gap changes sign and becomes negative at the end of the longlasting low-growth spell. This is due to the fact that convergence of capital to its perfectinformation level is relatively more sluggish than that of consumption. When the second shortlasting high-growth regime occurs, optimism is a bit smaller than in the previous high-growth
period, resulting in a more contained hike in the consumption gap. The dynamics of the
consumption gap and the capital gap are clearly reversed during the high-growth phase.
Let us focus on the accuracy of our method in tracking the dynamics of beliefs over time.
We initially set 400 equally spaced knots in our grid G. Furthermore, we add 100 knots to make
the grid …ner for beliefs near the convergence points for prob f t = 1j 1t g and prob f t = 3j 1t g,
which are zero in both cases. After the re…nement of the grid of beliefs introduced in steps
10-11 of Section 2.2.2, we are left with 152 grid points per block. Even if the number of regimes
b and
seems enormous, solving the model is fast. It takes 0.97 second to compute the matrix P
6.42 seconds to solve the model with gensys in Matlab on a 64-bit desktop endowed with an
27
Intel core processor i7-2600 CPU at 3.40 GHz.6 Figure 6 shows that the grid of beliefs …ts the
actual sequence of beliefs very accurately as squares and circles are almost perfectly concentric.
3.4
Sneaky Switches to Low-Growth Phase
So far we have studied cases in which transitions between high-growth and low-growth phases
are always marked by an observable switch in regimes. For instance, a switch to the lowgrowth phase is characterized by an observable change of TFP growth from high to low. This is
because, so far, we have assumed that the probability of switching between regimes belonging
to the same block is zero. In this section we relax this assumption.
Let us use the baseline calibration and the same values for the transition matrix P as those
used in Subsection 3.3, with the only exception being that now the probability of switching
to the short-lasting high-growth regime conditional on being in the long-lasting high-growth
regime is p21 = 0:04. The probabilities p23 and p24 are re-scaled so that (26) is satis…ed. In this
context, a switch from the high-growth phase to the low-growth phase may happen when the
economy is in the long-lasting high-growth regime. This means that the system could switch to
the low-growth phase while agents do not observe any switch from high growth to low growth.
Although the probability that such a sneaky switch would happen is quite small (p21 = 0:04),
such a possibility dramatically in‡uences the dynamics of agents’beliefs and allocations.
Figure 7 shows the dynamics of consumption and capital in the case of learning and perfect
information using the same sequence of regimes as that analyzed in the previous example.7 Let
us focus on the second half of the periods when the economy enters the high-growth phase. The
top left graph of Figure 7 shows that agents’beliefs about being in the long-lasting high-growth
regime do not converge to unity even when a large number of high-growth periods occur. This
is di¤erent from what we observe in Figure 5. Thus, an important implication of introducing
sneaky switches to the low-growth phase is that agents will never be fully convinced that they
are in the high-growth phase. Furthermore, as short-lasting low-growth regimes occur, agents
are relatively more concerned about the possibility of having entered a long-lasting low-growth
period. The reason is that agents are aware that a sneaky switch may have occurred during
the last high-growth regime.
The rightside graphs of Figure 7 show the consumption and capital gaps with respect to
the perfect-information benchmark. As we allow for the possibility of sneaky switches, the
high-growth phase is characterized by recurrent negative consumption gaps as the economy is
going through short-lasting low-growth regimes. This is di¤erent from the case of no sneaky
6
In the case with static prior beliefs, which was analyzed in Section 3.1, it takes 0.10 second to compute the
b and to solve the model with gensys.
matrix P
7
To ease the comparison with the previous case with no sneaky switches, the scale of the y-axes is the same
as that in Figure 5.
28
Beliefs
0.8
Consumption Gap
1
Pr. LL-HG
Pr. LL-LG
0.4
0.2
0.6
25
50
75
100
5
4
3
2
1
0
10
-5
-20
-35
25
50
75
100
25
50
75
100
20
4Q
8Q
20Q
40Q
Ergodi c
Capital Gap
E(Avg TFP Growth)
1
25
25
50
75
10
0
-10
-20
100
Figure 7: Sneaky Switches. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime
(red solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Annualized
percentage deviations of consumption from the perfect-information benchmark. Bottom left graph: Expected
average growth rate of technology (annualized percentage) at various horizons. Bottom right graph: Annualized
percentage deviations of capital from the perfect-information benchmark. In all graphs, gray areas denote
periods of low growth.
switches in Figure 5 in which we observe only one large negative consumption gap that gradually
disappears as the economy remains in the high-growth phase. The reason is that the possibility
of sneaky switches to the low-growth phase prompts households to interpret short-lasting lowgrowth regimes as long lasting. As a result, households adjust their consumption more than
their capital stock. When long-lasting high-growth periods occur, agents are initially not very
optimistic, expecting a quite short-lasting high-growth period. As a result, they speed up capital
accumulation to achieve consumption smoothing. Quite interestingly and unlike the example in
Figure 5, high pessimism during short-lasting low-growth periods causes the capital gap to not
exhibit mean reversion during a typical high-growth phase. Therefore, the possibility of sneaky
switches induces households to hoard capital during high-growth phases. Capital hoarding
during high-growth phases is due to households’inability to fully learn when the economy is in
the high-growth phase because of the possibility of sneaky switches to the low-growth phase.
Finally, in the low-growth phase, households learn faster that the economy is on a low-growth
path than in Figure 5. This translates into smaller departures of consumption and capital
allocations from the perfect-information benchmark.
29
3.5
Over-pessimism
So far, we have studied economies in which agents’pessimism or optimism monotonically increases as the system remains in the same block. In this section, we study the case in which
pessimism over-reacts as the system enters the low-growth block and gradually falls as the economy remains in the low-growth block. Let us consider the same parameterization as in Section
3.3. We make three departures from this case. First, the probability of switching from the longlasting low-growth regime to the short-lasting one is p43 = 0:15 > 0. Second, the probability of
staying in the long-lasting low-growth regime p44 is set to 0:80 < 0:95. Third, the probability
of switching from the long-lasting high-growth regime to the short-lasting low-growth regime
conditional on the system having switched to the low-growth block is
p23
= 0:25
p23 + p24
(27)
To sum up, the transition matrix reads:
2
0:75
6
6
0
P =6
6 0:0125
4
0:0475
0
0:95
0:0125
0:0125
0:2375
0:0025
0:75
0:15
3
0:2375
7
0:0375 7
7
7
0
5
0:80
Consider the following simulation in which all regimes occur with their expected durations.
First, starting from the highest optimism, the long-lasting high-growth regime is realized for 20
quarters followed by 4 quarters of the short-lasting low-growth regime. After that, the economy
switches again to the long-lasting high-growth regime and …nally to the long-lasting low-growth
regime for 5 quarters. Finally, the system switches to the short-lasting low-growth regime for
4 quarters.
The results are reported in Figure 8. During the …rst 20 quarters agents are correct in
believing that they are in the long-lasting high-growth regime. Therefore, consumption and
capital allocations are the same as those in the perfect-information benchmark. Upon the
switch to the short-lasting low-growth regime, we observe a hike in agents’pessimism captured
by a sharp increase in the probability attached to the long-lasting low-growth regime and a drop
in expected TFP growth at all horizons. The blue dashed line in the upper left graph shows
that the direction of learning goes from the long-lasting regime to the short-lasting regime, and
therefore, it is opposite to what is observed in all the previous examples. As the system stays
in the low-growth block, agents attach less probability to being in the long-lasting low-growth
regime. This is because from the long-lasting low-growth regime the economy can move to
the short-lasting low-growth regime. Agents are aware of this, and as time goes by, they take
30
Consumption Gap
1
0.4
0.2
0.6
E(Avg TFP Growth)
1
Pr. LL-HG
Pr. LL-LG
10
20
30
40
50
5
4
Capital Gap
Beliefs
0.8
3
2
4Q
8Q
20Q
40Q
Ergodi c
1
0
10
20
30
40
50
4
2
0
-2
-4
-6
2
1.5
1
0.5
0
-0.5
-1
10
20
30
40
50
10
20
30
40
50
Figure 8: Over-Pessimism. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime
(red solid line) and in the long-lasting low-growth regime (blue dashed line). Top right graph: Annualized
percentage deviations of consumption from the perfect-information benchmark. Bottom left graph: Expected
average growth rate of technology (annualized percentage) at various horizons. Bottom right graph: Annualized
percentage deviations of capital from the perfect-information benchmark. In all graphs, gray areas denote
periods of low growth. The vertical dashed red lines in all graphs mark the time at which the economy switches
from the long-lasting low-growth regime to the short-lasting low-growth regime.
into account the possibility that this event might in fact have occurred. Therefore, pessimism
initially overshoots and then falls as agents observe more and more periods of low growth. Such
over-pessimism has profound implications for consumption and capital allocations. The upper
right graph shows that the …rst hike in pessimism causes consumption to fall dramatically with
respect to the case of perfect information. This is because of a misalignment of households’
expectations. As agents’pessimism goes down, the consumption gap shrinks and the capital
gap widens.
When the economy switches to the long-lasting low-growth regime at t = 46, we observe
a second hike in pessimism. Yet, unlike in the previous low-growth period, households are
now correct in mostly expecting the current low-growth period to be long lasting. Thus, the
consumption and capital gaps do not exhibit any dramatic change upon the switch. In the subsequent periods, households’pessimism gradually declines and only then do the consumption
and capital gaps widen. This happens because uninformed agents take into account the possibility that the economy experienced a sneaky switch to the short-lasting low-growth regime,
while perfectly informed agents can observe that the long-lasting low-growth regime is still in
place. In the last four periods of low growth, the sneaky switch actually happens and it is
31
marked with a vertical red dashed line. Since this switch is hidden, it does not a¤ect households’beliefs and allocations in our economy, while in the counterfactual perfect information
economy it determines a drastic change in consumption and capital allocations that manifests
itself with a quick reversal in the dynamics of the consumption and capital gaps.
Summarizing, during low-growth periods the direction of learning is reversed with respect to
what we have seen in all previous examples. The longer the low-growth span, the less pessimistic
about the future duration of the low-growth period agents become. When the true regime is
long lasting, such a pattern of beliefs causes households to slow down capital accumulation
more aggressively than under perfect information. On the contrary, when the true regime is
short lasting, agents are initially too pessimistic and accumulate an extra stock of capital with
respect to what they would do if they knew the stochastic duration of the regime.
3.6
Bipolar Beliefs
In this section we provide a limiting case in which agents’beliefs change dramatically in every
period. We make the following two key assumptions: (i) high-growth regimes are assumed
to communicate a lot (p12 = 0:95, p21 = 0:65) and (ii) the relative persistence of low-growth
regimes is markedly di¤erent (p33 = 0:25 and p44 = 0:99). Furthermore, the probability of
staying in the high-growth regimes is p11 = 0 and p22 = 0:3. Low-growth regimes do not
communicate (p34 = p43 = 0). We use (25) and (26) for the probabilities linking regimes across
blocks. Condition (26), which implies that the persistent high-growth regime is associated with
a much smaller probability of switching to the very persistent low-growth regime, and condition
(ii) together imply that whether forward-looking households believe that they are in the shortlasting or in the long-lasting high-growth regime has large e¤ects on consumption and capital
allocations.
Consider a sequence of high-growth regimes that alternates one period of Regime 1 and one
period of Regime 2 ten times. The top left graph of Figure 9 reports the dynamics of agents’
beliefs about being in the long-lasting high-growth regime and the true probabilities are shown
in blue squares. Consistent with assumptions (25) and (26), we initialize agents’ beliefs to
attach a probability of 0.95 that the economy is in the short-lasting Regime 1 at time 1. Lots of
communication between regimes belonging to the same block brings about bipolar dynamics of
beliefs during the early quarters spent in the high-growth block. Such a pattern is exacerbated
by the fact that Regime 1 has zero persistence. As the system remains in the high-growth block,
bipolarism fades away because agents are not receiving any additional information. Therefore,
the probability that the high-growth regime has been in place for an even number of consecutive
periods keeps growing, breaking the bipolar dynamics.
The rightside graphs highlight the di¤erence between these allocations and the perfect-
32
Consumption Gap
0.8
0.6
0.4
0.2
0
Probabil ity LL-HG
True Probabil ity LL-HG
5
10
15
0.4
0.2
0
-0.2
20
5
10
15
20
5
10
15
20
0
3
Capital Gap
E(Avg TFP Growth)
Beliefs and True Prob.
1
4Q
8Q
20Q
40Q
5
10
15
-0.02
-0.04
-0.06
-0.08
20
Figure 9: Bipolar Beliefs. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime
(red solid line) and the true probability (blue squares) of being in the long-lasting high-growth regime. Top right
graph: Annualized percentage deviations of consumption from the perfect-information benchmark. Bottom left
graph: Expected average growth rate of technology (annualized percentage) at various horizons. Bottom right
graph: Annualized percentage deviations of capital from the perfect-information benchmark.
information benchmark. When the short-lasting Regime 1 is in place, perfectly informed agents
know that there is a fairly large probability that the economy will move to the highly persistent
low-growth Regime 4 if the system switches blocks. As a result, perfectly informed households
slow down the growth rate of consumption and raise that of capital, expecting the worst for
the future. Imperfectly informed households follow the same policy but much less aggressively,
bringing about a positive consumption gap. When the long-lasting Regime 2 is in place, perfectly informed households know that the most likely low-growth regime to occur in the next
period is the short-lasting one. Therefore, perfectly informed households reduce investment and
raise the growth rate of consumption. Imperfectly informed households are uncertain about
whether the regime is short or long lasting. Thus, they adjust capital less aggressively. This
leads to a negative consumption gap. Note that the lower left graph shows that when the
system has been in the high-growth block su¢ ciently long, the expected average growth rate
at every horizon does not change over time as beliefs start changing.
33
4
Signals and Implications for Uncertainty
Consider the following transition matrix:
2
0:85
6
6 0:05
P =6
6 0
4
0:1
0:1
0:9
0:025
0:05
0:99
0
0:01
0
3
0:025
7
0 7
7
0 7
5
0:99
While the high-growth regimes exhibit similar persistence, the low-growth regimes have markedly
di¤erent persistence. There is a very small probability that, once the system is in Regime 3,
it will stay in the low-growth block next period; most likely it will switch to the long-lasting
high-growth regime. Note that the probability of staying in the high-growth block is 0.95 for
both the short-lasting and long-lasting high-growth regimes. However, Regime 1 has a larger
downside risk with a bigger probability of moving to the long-lasting low-growth Regime 4.
When under the high-growth block, households receive a public signal $t about the regime
in place. The signal can take two values: 1 or 2. We assume that prob f$t = 1j t = 1g = 0:20
and prob f$t = 1j t = 2g = 0:80, implying that receiving a signal $t = 1 is more likely when the
economy is the short-lasting high-growth regime. Conversely, receiving a signal $t = 2 is more
likely when the economy is in the long-lasting high-growth regime. We study the evolution
of allocations and beliefs when Regime 2 is in place for its typical duration of 10 quarters.
Households always receive the same signal $t = 2 during the period except at time t = 3 and
t = 6, when they receive $t = 1. Figure 10 shows the dynamics of beliefs and allocations (black
dashed line) and compare them with those of an economy in which households always receive
$t = 2 at any time (solid blue line).
Receiving signals $3 = 1 and $6 = 1 in‡uences agents’beliefs by reducing their optimism.
Note that nothing is really changed in the economy’s fundamentals as the economy remains in
Regime 2 at all times. Hence, signals play the role of shocks to beliefs with the e¤ect of reducing
optimism. If households did not receive the signals $3 = 1 and $6 = 1, their beliefs would
have not changed (see the black dashed line). Such shocks to beliefs change consumption and
capital allocations. The …rst shock to beliefs reduces consumption by 0:85% and the second
one by 0:8% three periods later. Furthermore, Figure 10 shows that shocks to beliefs prompt
agents to accumulate more capital. The higher accumulated capital pushes consumption up at
the end of the simulated sample when the e¤ects of signals on beliefs has faded away.
Importantly, signals give rise to shocks to beliefs that have second-order e¤ects. In this
simulation, bad signals raise the expected probability of switching to the very persistent lowgrowth Regime 4, determining an increase in the downside risk and uncertainty. The bottom
middle panel and the bottom right panel show that the increase in downside risk translates
34
Figure 10: Belief Shocks. Top left graph: Evolution of beliefs of being in the long-lasting high-growth regime
in the case of shocks to beliefs at time t=3 and t=6 (black dashed line) and in the case of no shock to beliefs
(solid blue line). Top middle graph: Annualized percentage deviations of consumption from the case of no
shock to beliefs. Top right graph: Expected average growth rate of technology (annualized percentage points)
at various horizons (4 quarters, 8 quarters, 20 quarters, and 40 quarters) for the case of shocks to beliefs at t=3
and t=6. From top to bottom: the solid blue line denotes the horizon of 4 quarters, the solid black line denotes
the horizon 40 quarters. Bottom left graph: Annualized percentage deviations of capital from the case of no
shock to beliefs. Bottom middle graph: Uncertainty about future consumption at horizons from 4 quarters to
20 quarters. Lighter blue areas denote shorter horizons. Darker red areas denote longer horizons. Bottom right
graph: Uncertainty about future TFP growth at horizons from 4 quarters to 20 quarters.
into a spike in uncertainty about future consumption and future TFP growth. Note that such
changes in uncertainty are not supported by any changes in the economy’s fundamentals, but
rather are due to signals that change the perceived probability of switching to Regime 4.
While this is not the …rst paper to use signals as shocks to beliefs (e.g., Lorenzoni, 2009,
Angeletos and La’O, 2010 and Forthcoming) the approach proposed in this paper has the
important advantage of keeping the model very tractable. This feature makes our methods
suitable for studying shocks to beliefs in likelihood-based estimated large-scale DSGE models
(e.g., Christiano, Eichenbaum, and Evans, 2005 and Smets and Wouters, 2007).
35
5
Modeling Beliefs about Policy Makers
This section provides a concise review of two alternative applications of the methods described
in this paper.
5.1
Modeling Communication and Constrained Discretion
Bianchi and Melosi (2012a) apply the methods introduced in this paper to quantitatively study
monetary policy and the role central banks’communication in a prototypical New-Keynesian
DSGE model. In the model, monetary policy alternates between periods of active in‡ation
stabilization, in an active regime, and periods during which the emphasis on in‡ation stabilization is reduced, in a passive regime. Agents in the model are fully rational and able to infer if
monetary policy is active or not. However, when the passive regime prevails, they are uncertain
about the nature of the observed deviation. In other words, agents are not sure if the central
bank is engaging in a short-lasting or long-lasting deviation from active monetary policy.
Agents conduct Bayesian learning in order to infer the likely duration of the deviations from
active monetary policy. As agents observe more and more deviations, they become more and
more convinced that they are in the long-lasting passive regime and are increasingly pessimistic
about a quick return to the active regime. When the model is estimated to the U.S. economy,
we …nd that in‡ation expectations and agents’ uncertainty sluggishly increase as the Federal
Reserve keeps deviating from active policy and they accelerate only after 20 quarters of deviations. These features make the model well-suited to explaining the monetary policy framework
that has been adopted by the Federal Reserve, which Bernanke and Mishkin (1997) have termed
constrained discretion. Bernanke (2003) explains that under constrained discretion, the central
bank retains some ‡exibility in the conduct of monetary policy in order to accommodate shortrun disturbances. However, such ‡exibility is constrained to the extent that the central bank
should maintain strong credibility for keeping in‡ation and in‡ation expectations …rmly under
control.
Furthermore, Bianchi and Melosi (2012a) build on the methods introduced in this paper to
provide a very general framework to model policy-makers’communication. We study the case
(transparency) in which the central bank systematically announces the number of consecutive
deviations from active monetary policy. Since the model is purely forward-looking, a su¢ cient
statistic to solve the model with systematic announcements is the number of periods of announced passive policy that lie ahead before switching to the active policy. In other words,
the laws of motion associated with a situation in which the central bank announces …ve periods of consecutive deviations from the active regime are exactly the same as those associated
with a situation in which a central bank has carried out …ve consecutive deviations out of ten
announced deviations. Therefore, we can rede…ne the structure of regimes as follows: Regime
36
1 is the active regime; Regime 2 is a regime in which only one period of announced passive
policy (i.e., the current one) is left before switching to the active regime; Regime 3 is a regime
in which two consecutive periods of passive policy will be conducted before switching to the
active policy; etc. To avoid the possibility that the dimensionality of the set of regimes blows
up to in…nity, we truncate the duration of passive deviations to a . For any " > 0 we can …nd
a a such that the probability of a deviation longer than a has a probability smaller than ",
implying that the approximation can be made arbitrarily accurate.
Endowed with this result, we can study the impact of monetary policy communication on
welfare by rede…ning the structure of regimes in terms of the number of announced deviations
yet to be carried out a as follows:
(
a
t
= i) ;
y
(
a
t
= i) =
"
A
P
;
;
P
y
A
y
, if i = 1
, if 1 < i <
a
#
where A and A
y are the in‡ation and output gap coe¢ cient in the monetary policy reaction
function when monetary policy is active and P and Py when monetary policy is passive. The
eA , which
re-de…ned set of regimes at are governed by the ( a + 1) ( a + 1) transition matrix P
is de…ned as:
"
#
p
p
e
11
A
eA =
P
Ia 0a 1
where p11 is the probability of staying in the active regime, I a is a a
a identity matrix,
0 a 1 is a ( a 1) column vector of zeros. and peA is a 1
a row vector whose typical ith element is the probability of announcing i consecutive periods of passive monetary policy
followed by a switch to the active regime, conditional on being in the active regime. Formally,
Pa 1
eA (i). Note that
1 and peA ( a ) = 1 p11
peA (i) p12 pi22 p21 + p13 pi33 p31 for 1 i
a
i=1 p
eA is a function of the probabilities of the transition matrix P for the primitive
the matrix P
regimes.
5.2
Dormant Fiscal Shocks
Bianchi and Melosi (forthcoming) develop a model in which the current behavior of the …scal
and monetary authorities in‡uences agents’beliefs about the way debt will be stabilized. The
standard policy mix consists of a virtuous …scal authority that moves taxes in response to debt
and a central bank that has full control over in‡ation. When policy-makers deviate from this
virtuous policy mix, agents conduct Bayesian learning to infer the likely duration of the deviation. As agents observe more and more deviations, they become increasingly pessimistic about
a prompt return to the virtuous regime and in‡ation starts moving to keep debt on a stable
path. Shocks that were dormant under the virtuous policy mix now start to manifest themselves
37
and uncertainty about the macroeconomy starts increasing among agents. These changes are
initially imperceptible, but they unfold over decades and accelerate as agents become convinced
that the …scal authority will not raise taxes. Dormant …scal shocks can account for the run-up
of in‡ation in the ‘70s and the de‡ationary pressure of the early 2000s. The paper also shows
that the currently low long-term interest rates and in‡ation expectations might hide the true
risk of in‡ation faced by the US economy.
Bianchi and Melosi (2012b) construct an economy in which policy-makers’reputation for …scal virtue smoothly ‡uctuates over time. The monetary and …scal policy mix alternates between
periods of …scal virtue and periods of …scal irresponsibility. Under …scal virtue, a virtuous rule
is in place most of the time: the central bank stabilizes in‡ation and the government strongly
adjusts taxes to stabilize public debt. Under …scal irresponsibility, a …scally irresponsible rule is
in place most of the time: the central bank de-emphasizes in‡ation stabilization and the …scal
authority is not committed to keeping debt under control. A strong reputation for …scal virtue
is generally desirable because it leads to a stable macroeconomic environment, but when the
economy enters the zero lower bound, policy-makers face a trade-o¤ between preserving their
reputation and escaping a large recession. Given that policy-makers’behavior is constrained
at the zero lower bound, beliefs about the exit strategy are substantially more important than
policies implemented during the crisis. Announcing a period of austerity is detrimental in the
short run, but it preserves macroeconomic stability in the long run. A severe recession can
be avoided by abandoning …scal virtue, but this results in a sharp increase in macroeconomic
instability. Contradictory announcements by the …scal and monetary authorities can lead to
high in‡ation and large output losses. Finally, the paper shows that high uncertainty is an
inherent implication of entering the zero lower bound while de‡ation is not, because agents are
likely to be uncertain about the way policy-makers will deal with the large stock of debt arising
from a severe recession.
6
Concluding Remarks
This paper has developed methods to solve general equilibrium models in which agents are
subject to waves of optimism, pessimism, and uncertainty. Agents in the model are fully
rational, understand the structure of the economy, and know that they do not know. Therefore,
when forming expectations they take into account that their beliefs will evolve in response to
realized observable economic outcomes, the behavior of other agents in the model, or both.
The central insight consists of creating an expanded number of regimes indexed with respect to
agents’beliefs. The resulting law of motion re‡ects agents’uncertainty and can be expressed
in state space form. Therefore, the framework proposed in this paper is suitable for structural
estimation.
38
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41
Appendices
The appendices are organized as follows. Appendix A works out the recursions (3) and (4)
that pin down the dynamics of beliefs within blocks. Appendices B-F prove Propositions 26. Appendix G shows that the unstable root e1 is either smaller than or equal to zero or
higher than or equal to unity. Appendix H details the algorithm to construct the transition
b when agents receive signals. Appendix I characterizes the steady-state equilibrium for
matrix P
stationary variables in the RBC model and obtains the log-linearized equations of this model.
Note that the convergence results, which are proven in Appendices B-G, could be derived by
working on the submatrices of each block. However, we have decided to work with the solution
of the di¤erence equations (3) and (4) because this approach is familiar to a wider audience.
A
Deriving the Law of Motion for Beliefs
In this appendix, we want to show two propositions.
Proposition 7 The rational di¤ erence equations (3) and (4) hold true
Proof. Recall that equation (3) describes the dynamics of beliefs within Block 1. Consequently, this
equation holds when 1t > 1. The Bayes’ theorem can be applied to characterize the probability of
being in Regime 1 given that the system is in Block 1 ( 1t > 1):
prob
But if
1
t
=
1
t 1
t
1
t
= 1j
1
t
p
= P4
=
1
t
i=1 p
1
t 1
+ 1j
=
1
t 1
t
=1 p
+ 1j
t
t
=i p
= 1j
t
=
1
t 1
ij 1t 1
+ 1, then the likelihood is such that
p
1
t
=
1
t 1
+ 1j
t
=1 =p
1
t
=
1
t 1
+ 1j
t
=2 >0
p
1
t
=
1
t 1
+ 1j
t
=3 =p
1
t
=
1
t 1
+ 1j
t
=4 =0
and
The equality in the …rst expression re‡ects the fact that agents cannot distinguish regimes belonging
to the same block. The inequality sign in the …rst expression and the equality sign in the second
expression are due to the fact that the system is in Block 1 at time t, ruling out the possibility that
either Regime 3 or Regime 4 is realized. These results allow us to write:
prob
Since p
t
= ij
1
t 1
=
P2
j=1 p
prob
t 1
t
t
= 1j
= jj
= 1j
1
t
1
t
1
t 1
p
= P2
t
i=1 p
= 1j
t
=
1
t 1
ij 1t 1
pji , then
P2
1
t 1 = jj t 1 pj1
j=1 p
P2
1
t 1 = jj t 1 pji
i=1
j=1 p
= P2
Furthermore, note that p t 1 = 2j 1t 1 = 1 p t 1 = 1j 1t 1 and after straightforward manipulations leads to equation (3). Equation (4) can be proved analogously.
42
B
Proof of Proposition 2
If (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 6= p22 or both p12 6= 0 and p21 6= 0, and
the initial probability is such that prob t = 1j 1t = 1 6= e1 ; then prob t = 1j 1t
! e2 2 [0; 1]. If
conditions (i), (ii), and (iii) hold and the initial probability is such that prob t = 1j 1t = 1 = e1 ,
then prob t = 1j 1t = e1 for any 1t .
The di¤erence equation (3) can be expressed as
prob
t
= 1j
1
t
=
a prob
c prob
t 1
t 1
= 1j
= 1j
1
t 1
1
t 1
+b
+d
(28)
where
a
p11
p21 ; b
c
p11 + p12
p21
p21
p22 ; d
p21 + p22
Condition (i) ensures that the di¤erence equation of interest is rational because it implies c > 0.
In Appendices E and F, we will deal with the case of c = 0. We then proceed as follows. Denote
prob t = 1j 1t + dc as xt and re-write the di¤erence equation above as
xt =
xt
(29)
1
where
p11 + p22
p11 + p12 p21 p22
p11 p22 p21 p12
(p11 + p12 p21 p22 )2
Condition (ii) ensures that 6= 0. The case of = 0 will be studied in Appendix D. The above
equation can be reduced to a homogeneous linear di¤erence equation by de…ning xt = 't ='t 1 where:
't
't
1
+ 't
=0
2
(30)
1
2
If 1 and 2 are the solutions of the characteristic equation, namely
solution of (30) is
't = C1
t
1
+ C2
't = (C1 + C2 t)
t
2;
t
1;
if
1
if
1
6=
=
2
2
1
2
p
2
4 , then the general
(31)
(32)
The general solution of (29) is then:
xt =
when C2 = 0, xt =
1
t
1
C1 t1 1
C1
for all t. When C1 = 0, xt =
2
t+1
xt =
1
2
2
t
1
2
+ C2
+ C2
(33)
for all t. When neither C1 nor C2 is zero, then
+C
+C
43
t
2
t 1
2
; C 6= 0
(34)
Note that
2
4 is required for the characteristic roots
2
p11 + p22
p11 + p12 p21
4
p22
1
and
p11 p22
(p11 + p12
2
to be real. This condition is
p21 p12
p21 p22 )2
and after simplifying
p211 + p222 + 2p11 p22
4p11 p22
4p21 p12
Some straightforward manipulation leads us to
(p11
p22 )2
4p21 p12
(35)
From condition (iii ), the inequality above is strict and the characteristic roots are unequal. The
case in which the characteristic roots are identical is dealt with in Appendix C. Let j 2 j > j 1 j
then j 1 = 2 jt ! 0 and (34) implies that
p xt ! 2 as long as x1 6= 1 . The root with highest absolute
p +p +
(p
p )2 +4p p
22
11
22
21 12
value can be seen to be always 11 2(p
. Recall that xt prob t = 1j
11 +p12 p21 p22 )
some straightforward algebraic manipulations we obtain:
q
p
p
2p
+
(p11 p22 )2 + 4p21 p12
11
22
21
prob t = 1j 1t ! e2 =
2 (p11 + p12 p21 p22 )
where e2 is the stable root for the variable of interest prob
prob t = 1j 1t can be easily seen to be:
e1 =
p11
p22
2p21
q
(p11
2 (p11 + p12
t
= 1j
p21
p21
.
+ dc . After
The unstable root for
p22 )2 + 4p21 p12
p22 )
We only need to show that e2 2 [0; 1]. We want to show that
q
p11 p22 2p21 + (p11 p22 )2 + 4p21 p12
2 (p11 + p12
1
t
1
t
p22 )
0
If p11 + p12 p21 p22 > 0 and p11 p22 2p21
0, then the statement is clearly true. When
p11 + p12 p21 p22 > 0 and p11 p22 2p21 < 0, then
q
(p11 p22 )2 + 4p21 p12
(p11 p22 2p21 )
Since the right-hand side is positive we can square both sides of this equation:
(p11
p22 )2 + 4p21 p12
(p11
p22
4p21 p12
4p221
4 (p11
If p21 = 0, the statement is true. If p21 > 0
p12
p21 + (p11
which is true.
44
p22 )
0
2p21 )2
p22 ) p21
If p11 + p12
p21
p22 < 0, then p11
p11
p22
p22
2p21 < 0. We need to show that
q
(p11 p22 )2 + 4p21 p12
2p21
Since both sides of the inequality are negative, then
(p11
2p21 )2
p22
p22 )2 + 4p21 p12
(p11
and after manipulating:
p22 ) p21 + 4p221
4 (p11
4p21 p12
If p21 = 0, the inequality is obviously veri…ed. If p21 > 0, then
0
(p11
p22 ) + p12
p21
which is true.
We want to show that
p11
p22
2p21 +
q
(p11
2 (p11 + p12
If p11 + p12
p21
p22 )2 + 4p21 p12
p21
p22 > 0, then after some manipulations
q
(p11 p22 )2 + 4p21 p12 p11 + 2p12
Note that p11 + 2p12
inequality yields:
p22 > p11 + p12
(p11
p21
1
p22 )
p22
p22 > 0. Hence, taking the square on both sides of the
p22 )2 + 4p21 p12
(p11 + 2p12
p22 )2
and …nally
4p21 p12
4p212 + 4 (p11
p22 ) p12
If p12 = 0, this is true. If p12 > 0, then
p21
which is true.
If p11 + p12
If p11 + 2p12
p21
p12 + (p11
p22 )
p22 < 0, then after some manipulations
q
(p11 p22 )2 + 4p21 p12 p11 + 2p12
p22
p22 < 0, this inequality is obviously true. If p11 + 2p12
(p11
p22 )2 + 4p21 p12
(p11 + 2p12
and then
4p21 p12
4p212 + 4 (p11
p22 ) p12
If p12 = 0, this is true. If p12 > 0, then
p21
p12 + p11
which is true.
45
p22
p22
p22 )2
0, then
C
Proof of Proposition 3
We want to show that if (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 6= p21 p12 , (iii) p11 = p22 and either
p12 = 0 or p21 = 0, then prob t = 1j 1t ! e1 = e2 and the roots are either equal to zero (if p21 = 0)
or one (if p12 = 0). This result follows from observing that condition (iii) implies that condition (35)
delivers coincident characteristic roots e1 and e2 ; that is,
If p12 = 0, then prob
D
t
= 1j
1
t
e1 = e2 =
p21
p21 p12
! e1 = e2 = 1. If p21 = 0, then prob
t
= 1j
Proof of Proposition 4
1
t
! e1 = e2 = 0.
We want to show that if (i) p11 + p12 p21 p22 6= 0, (ii) p11 p22 = p21 p12 , then prob t = 1j 1t =
p11 p21
= 0 in equation (29) and hence (using the notation introduced
p11 +p12 p21 p22 . Condition (ii) implies
in Appendix B)
p11 + p22
xt =
p11 + p12 p21 p22
From Appendix B, recall that xt = prob
prob
E
t
t
= 1j
= 1j
1
t
1
t
p11
p11 + p12
=
+ d=c, then it follows that
p21
p21
p22
.
Proof of Proposition 5
We want to show that if (i) p11 + p12 p21 p22 = 0 and (ii) p11 6= p21 , then prob t = 1j 1t !
p21
p21
p21 p22 = 0, then c = 0 in the di¤erence
p22 p11 +2p21 , with p22 p11 +2p21 2 [0; 1] : If p11 + p12
equation (28), which hence boils down to the …rst-order linear di¤erence equation below:
prob
t
= 1j
1
t
=
a
prob
d
t 1
= 1j
1
t 1
+
b
d
(36)
11 p21
where a = p11 p21 ; b = p21 ; d = p21 + p22 . Stability is ensured by ad = pp21
+p22 < 1. First note
that the background assumption A1 combined with condition (i) implies that d 6= 0 and hence the
ratio ad is well-de…ned. Condition (ii) rules out the possibility that the ratio ad is zero. We consider
this case in Appendix F.
p21
The condition p11 +p12 p21 p22 = 0 allows us to re-write the stability condition ad = pp11
as
21 +p22
p11 p21
p11 +p12
. Hence, showing that p12 + p21 > 0 implies stability. Recall that the background assumption
A2 requires that either p11 6= p22 or p12 6= p21 . If the latter condition is satis…ed, then p12 + p21 > 0
trivially follows. If the latter condition is not satis…ed, then it must be that p11 6= p22 , which, combined
with condition (i), implies that p12 + p21 > 0.
1
It is easy to see that the di¤erence equation (36) implies that prob t = 1j 1t ! db 1 ad
, that
is,
p21
p11 p21 1
prob t = 1j 1t !
1
p21 + p22
p21 + p22
46
After easy algebraic manipulations
prob
t
= 1j
1
t
Note that
0
!
p21
:
p11 + 2p21
p22
p21
p11 + 2p21
p22
1
To see that, recall that in this case, p11 + p12 p21 p22 = 0, implying that p22
Substituting this result into the inequalities above yields
p21
p12 + p21
0
p11 = p12
p21 .
1
which is clearly veri…ed.
F
Proof of Proposition 6
21
We want to show that if (i) p11 + p12 p21 p22 = 0, (ii) p11 = p21 , then prob t = 1j 1t = p22p+p
.
21
Condition (i) implies that c = 0 in the di¤erence equation (28), which hence boils down to the …rstorder linear di¤erence equation below:
prob
t
= 1j
1
t
=
a
prob
d
where a = p11
p21 ; b = p21 ; d = p21 + p22 .
prob t = 1j 1t = b=d = p21 = (p21 + p22 ).
G
t 1
= 1j
1
t 1
+
b
d
(37)
Condition (ii) implies that a = 0 and hence
Admissible Region for the Unstable Root
Recall that
We want to show that 0
e1
Lemma 8 If p11 + p12
p21
p11
p22
2p21
e1
p21
p22 )
1. This claim is implied by the following two Lemmas.
p22 > 0, then e1
p11
p22
2p21
0.
q
(p11
2 (p11 + p12
p21
p22 )2 + 4p21 p12
(p11
2 (p11 + p12
Proof. We want to show that
If p11 + p12
q
p21
p22 > 0, then the above implies
q
p11 p22 2p21
(p11
p22 )2 + 4p21 p12
p22 )
0
p22 )2 + 4p21 p12
Note that the background assumption A3 excludes that p11 p22 2p21 = 0. Hence there are two
possible cases left: (a) if p11 p22 2p21 < 0, then the above is true; (b) if p11 p22 2p21 > 0, then
we can take the square on both sides of the above equation to get
(p11
p22
2p21 )2
(p11
47
p22 )2 + 4p21 p12
Straightforward manipulations lead to
p221
p11 p21 + p22 p21
p21 p12
If p21 = 0, then the above is true. Otherwise, we can divide both sides of the above inequality by p21
to get
p11 + p12 p21 p22 0
that is obviously true because p11 + p12
Lemma 9 If p11 + p12
p21
Proof. We want to show that
p21
p22 > 0.
p22 < 0, then e1
p11
p22
2p21
1:
q
(p11
2 (p11 + p12
Since p11 + p12
p21
p11
and after simplifying
p22 )2 + 4p21 p12
p21
p22 < 0, the above implies
q
p22 2p21
(p11 p22 )2 + 4p21 p12
q
(p11
1
p22 )
p22 )2 + 4p21 p12
p11
2 (p11 + p12
p21
p22 )
p22 + 2p12
Note that the background assumption A3 excludes that p11 p22 + 2p21 = 0. If p11 p22 + 2p12 > 0,
the above is obviously true. If p11 p22 + 2p12 < 0, then taking the square on both sides
(p11
p22 )2 + 4p21 p12
p22 + 2p12 )2
(p11
After some manipulations:
p12 + p11
that is obviously true because p11 + p12
H
p21
p12
p22
0
p22 < 0.
Algorithm for the Case with Signals
b = 0g
Algorithm Set i = 1 and initialize the matrix P
Step 1 Find j1
where
prob
t
g1 and j2
= 1jIt ; $t
1
g1 so as to min
prob
g
t
= 1jIt ; $t
1; $
t
=q
Gjq with q 2 f1; 2g
prob ($t = qj t = 1) prob t = 1jIt ; $t 1
; $ t = q = P2
t
j=1 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
(38)
and agents’beliefs about being in Regime 1 before observing the signal read:
prob
t
= 1jIt ; $t
1
=
prob
t
prob t
1 = 1jIt
= 1jIt 1 ; $t 1 (p11 p21 ) + p21
t 1 (p + p
p21 p22 ) + p21 + p22
1; $
11
12
1
(39)
using the approximation prob t 1 = 1jIt 1 ; $t 1 = Gi . To ensure convergence of beliefs, we
correct j1 and j2 as follows. If jq = i and Gi 6= e2 (q 2 f1; 2g), then set jq = jq + 1 if Gi < e2
and jq = max (1; jq 1) if Gi > e2 .
48
Step 2 Setting prob t 1 = 1jIt 1 ; $t
as:
b (i; jq ) = P2 prob
P
v=1
1
t
= Gi , the (ex-ante) transition probability can be computed
= vjIt
where
prob
t
= vjIt
1; $
t 1
=
1; $
t 1
P2
u=1 prob
Step 3 Find j1 > g1 and j2 > g1 so as to min prob
where
prob
t
= 3jIt ; $t
1
prob f$t = qj
t
t 1
t
= ujIt
= 3jIt ; $t
1; $
= vg ; q 2 f1; 2g
1; $
t
t 1
(40)
puv
(41)
Gjq with q 2 f1; 2g,
=q
prob ($t = qj t = 3) prob t = 3jIt ; $t 1
; $ t = q = P4
t
j=3 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and the beliefs about being in Regime 3 upon the shift to Block 2 (before having observed the
signal $t ) are given by:
P
t 1 p
j3
t 1 = jjIt 1 ; $
j2b1 prob
t 1
P
prob t = 3jIt ; $
=P
t
1
pji
t 1 = jjIt 1 ; $
i2b2
j2b1 prob
=
prob
prob t 1 = 1jIt 1 ; $t 1 p13 + 1
t 1 (p + p ) + 1
13
14
t 1 = 1jIt 1 ; $
using the approximation that prob t 1 = 1jIt
Gi , the (ex-ante) transition probabilities as
b (i; jq ) = P
b (i; jq ) +
P
4
2
X
X
v=3
prob
t 1
1; $
= ujIt
prob
prob
t 1
1; $
t 1
t 1
t 1
= 1jIt
= 1jIt
1; $
1
puv
u=1
p23
(p23 + p24 )
; $t 1
= Gi . Setting prob
!
t 1
t 1
prob f$t = qj
t
= 1jIt
1; $
t 1
= vg ; q 2 f1; 2g
(42)
Step 4 If i = g1 then set i = i + 1 and go to step 6; otherwise, set i = i + 1 and go to step 1.
Step 5 Find j1 > g1 and j2 > g1 so as to min
where
prob
t
= 3jIt ; $t
1
prob
t
= 3jIt ; $t
1; $
t
=q
Gjq with q 2 f1; 2g
prob ($t = qj t = 3) prob t = 3jIt ; $t 1
; $ t = q = P4
t
j=3 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and agents’beliefs about being in Regime 3 before observing the signal read:
prob
t
= 3jIt ; $t
1
=
prob
t
prob t
1 = 3jIt
= 3jIt 1 ; $t 1 (p33 p43 ) + p43
t 1 (p + p
p43 p44 ) + p43 + p44
1; $
33
34
1
(43)
using the approximation prob t 1 = 3jIt 1 ; $t 1 = Gi . To ensure convergence of beliefs, we
correct j1 and j2 as follows. If jq = i and Gi 6= e4 (q 2 f1; 2g), then set jq = min (jq + 1; g) if
Gi < e4 and jq = jq 1 if Gi > e4 .
Step 6 Setting prob
as:
t 1
b (i; jq ) = P
b (i; jq ) +
P
= 3jIt
1; $
4
4
X
X
v=3
t 1
prob
= Gi , the (ex-ante) transition probability can be computed
t 1
= ujIt
u=3
1; $
t 1
puv
!
prob f$t = qj
t
= vg ; q 2 f1; 2g
(44)
49
=
Step 7 Find j1
where
prob
t
g1 so as to min prob
g1 and j2
= 1jIt ; $t
1
t
= 1jIt ; $t
1; $
t
=q
Gjq with q 2 f1; 2g,
prob ($t = qj t = 1) prob t = 1jIt ; $t 1
; $ t = q = P2
t
j=1 prob ($ t = qj t = j) prob ( t = jjIt ; $
1)
; q 2 f1; 2g
and the beliefs about being in Regime 1 upon the shift to Block 1 (before having observed the
signal $t ) are given by:
P
t 1 p
j1
t 1 = jjIt 1 ; $
j2b2 prob
t 1
P
P
prob t = 1jIt ; $
=
t
1
pji
t 1 = jjIt 1 ; $
i2b1
j2b2 prob
=
prob
prob t 1 = 3jIt 1 ; $t 1 p31 + 1
t 1 (p + p ) + 1
31
32
t 1 = 3jIt 1 ; $
prob
prob
t 1
t 1
= 3jIt
= 3jIt
1; $
1
t 1
; $t 1
p41
(p41 + p42 )
using the approximation that prob t 1 = 3jIt 1 ; $t 1 = Gg1 +i . Setting prob t 1 = 3jIt 1 ; $t
Gi , the (ex-ante) transition probability can be computed as:
!
2
4
X
X
t
1
b (i; jq ) = P
b (i; jq ) +
puv prob f$t = qj t = vg ; q 2 f1; 2g
P
prob t 1 = ujIt 1 ; $
v=1
u=3
(45)
Step 8 If i = g, then go to step 9; otherwise, set i = i + 1 and go to step 5.
b has all zero elements, then stop. Otherwise, go to step 10.
Step 9 If no column of P
P
b (i; j) 6= 0
Step 10 Construct the matrix T as follows. Set j = 1 and l = 1. While j
g, if gi=1 P
then do three things: (1) set T (j; l) = 1, (2) set T (j;
= 0 for any 1 v g and v 6= l, (3)
Pv)
g
b (i; j) = 0), set j = j + 1.
set l = l + 1 and (4) set j = j + 1; otherwise (i.e., if i=1 P
bR = T P
b T 0 . If no column of P
bR has all zero elements,
Step 11 Write the transition equation as P
R
b
b
set P = P and stop. Otherwise, go to step 10.
I
Log-Linearization of the RBC Model
Solving the problem of the representative household in Section 3 leads to:
ct
1
=
et c 1
E
t+1
ct + kt = zt kt
1
1
zt+1 kt
+ (1
) kt
+1
(46)
(47)
1
The stochastic process of TFP (19) and equations (46)-(47) imply that consumption and capital are
(1
) 1 e
(1
) 1
non-stationary. Denote the stationary variables e
ct ct =zt
, kt kt =zt
, t ln (zt =zt 1 ),
and Mt zt =zt 1 as the gross growth rate of TFP. The stationary version of the model reads:
e
ct
1
1
=
1
et e
E
ct+1
Mt+11
e
ct + e
kt = M e
kt
1
+ (1
h
Mt+1 e
kt
1
) Mt
1
e
kt
1
1
+1
i
(48)
(49)
Following Schorfheide (2005) and Liu, Waggoner, and Zha (2011), we de…ne a steady-state equilibrium for the stationary consumption e
ct and capital e
kt when "t = 0 all t and the growth rate of TFP
50
1
=
is at its ergodic value . The steady-state equilibrium level of consumption css and capital kss is:
2
0
1 @M
= 4
M
kss
css = M
1
13
1
1
h
kss + (1
1 + A5
1
)M
1
1
1
(50)
i
1 kss
(51)
where M exp ( ),
(p1 + p2 ) H + (p3 + p4 ) L is the ergodic mean of the log growth rate of the
economy, and pi stands for the ergodic probability of being in Regime i.
Taking the log-linear approximation of equations (48)-(49) around the steady-state equilibrium
(50)-(51) leads to
1
et b
b
ct = E
ct+1
(
css b
ct + kss b
kt =
M
1) 1 + (
1) M
b
kt
1
1
1
1
1
+ M
1
(
et bt+1
1) + 1 E
= 1 from equation (50) and bt
where we use the fact that M 1 M kss 1 + 1
is the
t
log-deviation of the growth rate of TFP from its ergodic mean . b
ct and b
kt denote log-deviations of the
stationary consumption and capital, respectively, from their steady-state value, and b ( t )
t ( t)
is the log-deviation of the TFP drift from its ergodic mean . The resource constraint is
1
kss
1
+
1
1
1
M
1
1
kss bt + M
1
kss + (1
and the log-deviations of the growth rate of TFP from its ergodic level follows
bt = bt ( t ) +
51
z "t :
)M
1
kss b
kt
1
(52)